I don't believe they're interchangeable. In the gospels 'son of God' is a term that others use for Yeshua (except for two instances), and 'the son of man' is a term that only Yeshua uses (except for one instance).
(Note: throughout this answer I will be quoting the Authorized KJV. When the KJV has added a definite article where there is none in the Greek manuscripts, I will replace the entire verse with a literal translation and add the Greek of the verse I changed)
Son of God
Since Matthew, Mark, and Luke are so similar, I'm only going to cover Matthew and John. I do find it interesting that 'son of God' is found in Mark only 3-4 times and 6-7 times in Luke.
In Matthew, there are nine occurrences. The first three are from the tempter and unclean spirits, so that leaves six. The first time Yeshua is called 'a son of God' by men is Matthew 14:33:
"those moreover in the boat bowed down to him saying truly of God a son you are" (Literal)
οἱ δὲ ἐν τῷ πλοίῳ προσεκύνησαν αὐτῷ λέγοντες· ἀληθῶς θεοῦ υἱὸς εἶ
The first time Yeshua is called 'the son of God' is Matthew 16:16:
"And Simon Peter answered and said, Thou art the Christ, the son of the living God." (KJV; both Mark and Luke exclude 'the son of the living God)
The next occurrence is pretty interesting:
"But Jesus held his peace. And the high priest answered and said unto him, I adjure thee by the living God, that thou tell us whether thou be the Christ, the son of God.
Jesus saith unto him, Thou hast said: nevertheless I say unto you, Hereafter shall ye see the son of man sitting on the right hand of power, and coming in the clouds of heaven." (KJV) Matthew 26:63
In this verse, the priest asked Yeshua if he was the son of God. Yeshua's only response is "you have said", and then he begins talking about 'the son of man'.
The next two occurrences are men mocking Yeshua before his death, and the final occurrence says:
"The moreover centurion and those with him keeping guard of the Yeshua having seen the earthquake and the things taking place feared greatly saying Truly of God a son was he." (Literal) Matthew 27:54
ὁ δὲ ἑκατοντάρχης καὶ οἱ μετ’ αὐτοῦ τηροῦντες τὸν Ἰησοῦν ἰδόντες τὸν σεισμὸν καὶ τὰ γινόμενα ἐφοβήθησαν σφόδρα, λέγοντες· ἀληθῶς θεοῦ υἱὸς ἦν οὗτος
John uses this term a lot. Since we've already seen that others call Yeshua 'son of God', I'll just focus on the few times Yeshua says it. In chapter 5, the Jews are angry because Yeshua calls God his father. They think this means Yeshua is making himself equal to God, but Yeshua says:
"Verily, verily, I say unto you, The son can do nothing of himself, but what he seeth the father do: for what things soever he doeth, these also doeth the son likewise." (KJV) John 5:19
He then says:
"Verily, verily, I say unto you, The hour is coming, and now is, when the dead shall hear the voice of the son of God: and they that hear shall live.
For as the Father hath life in himself; so hath he given to the son to have life in himself; (KJV)
And authority he gave to him judgement to execute [also], because he is a son of man." (Literal) John 5:25-27
καὶ ἐξουσίαν ἔδωκεν αὐτῷ κρίσιν ποιεῖν, ὅτι υἱὸς ἀνθρώπου ἐστίν
In this passage, Yeshua says that 'the dead' will hear the voice of 'the son of God'. He is obviously speaking figuratively here, because dead people cannot hear anything. From what I can gather, this correlates with:
Jesus said unto her, I am the resurrection, and the life: he that believeth in me, though he were dead, yet shall he live" (KJV) John 11:25
"And thou shalt say unto Pharaoh, Thus saith YHVH, Israel [is] my son, [even] my firstborn" (KJV) Exodus 4:22
The next time Yeshua says 'son of God' is during a similar scenario, where the Jewish leaders are upset because they somehow think calling oneself 'son of God' means they're calling themselves God:
The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.
Jesus answered them, Is it not written in your law, I said, Ye are gods?
If he called them gods, unto whom the word of God came, and the scripture cannot be broken; (KJV)
whom the father made set-apart and sent into the world you do say, 'because you blasphemy' because I said 'a son of the God I am?'" (Literal) John 10:33-36
ὃν ὁ πατὴρ ἡγίασεν καὶ ἀπέστειλεν εἰς τὸν κόσμον ὑμεῖς λέγετε ὅτι βλασφημεῖς ὅτι εἶπον υἱὸς θεοῦ εἰμι
Yeshua is trying to get his accusers to explain why saying "I am a son of God" is so bad. He points out that other men are explicity called "gods" and all he has said is "I am a son of God".
Now we are the sons of God
The reason Yeshua is a son of God is because God has many more children. John, who says son of God more than anyone else, tells us:
"But as many as received him, to them gave he power to become the sons [children] of God, [even] to them that believe on his name" John 1:12 (KJV)
And if the author of 1 John is the same:
"Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is." 1 John 3:2 (KJV)
So this is not a term that applies to Yeshua alone, nor when he says it can it be inferred that he only means himself. Those who follow God's word and believe Yeshua is the Messiah are "sons of God".
The son of man
The son of man is a different term that only Yeshua uses in the gospels, except for one instance. Many believe Yeshua was referring to Daniel, where it says "I saw one like a son of man". The only verses in the NT that imply this has anything to do with Yeshua are Rev. 1:13 and Rev. 14:14. Revelation borrows quite a bit from Daniel, and it's covered with symbolic and figurative language, so to conclude anything from this book alone probably won't help us much.
Yeshua uses this term differently than it's used in the Scriptures. He says ὁ υἱὸς τοῦ ἀνθρώπου which is literally 'the son of the man'. The closest Scripture to this is:
"What is man that you are mindful? And a son of man that you visit?" Psalm 8:4 (Literal)
מה־אנוש כי־תזכרנו ובן־אדם כי תפקדנו
This verse is used in the book of Hebrews, and the author is quite aware that this passage applies to mankind.
"For unto the angels hath he not put in subjection the world to come, whereof we speak. (KJV)
Has testified moreover somewhere someone, What is man, that you are mindful of him? or a son of man, that you visit him? (Literal)
διαμαρτύρομαι δέ πού τὶς λέγω τίς εἰμί ἄνθρωπος ὅτι μιμνήσκω αὐτός ἤ υἱός ἄνθρωπος ὅτι ἐπισκέπτομαι αὐτός
Thou madest him a little lower than the angels; thou crownedst him with glory and honour, and didst set him over the works of thy hands:
Thou hast put all things in subjection under his feet. For in that he put all in subjection under him, he left nothing that is not put under him. But now we see not yet all things put under him." (KJV) Hebrews 2:5-8
The author says that God has made all things for mankind and has given man authority over the works of His hands. However, we don't yet see this being used to its full potential-
"But we see Jesus, who was made a little lower than the angels for the suffering of death, crowned with glory and honour; that he by the grace of God should taste death for every man." (KJV) Hebrews 2:9
Yeshua is the son of man because he is a man, just like we are all the son of man.
Since this term is used very frequently, I'll only cover a few that I find significant.
The son of man can forgive sins
In Matthew, Yeshua tells a paralytic that his sins are forgiven. The Jewish leaders think this is blasphemy because they've decided in their own heads that only God can forgive sins. But Yeshua says:
"Wherefore think ye evil in your hearts?
For whether is easier, to say, Thy sins be forgiven thee; or to say, Arise, and walk?
But that ye may know that the son of man hath power on earth to forgive sins, (then saith he to the sick of the palsy,) Arise, take up thy bed, and go unto thine house.
And he arose, and departed to his house.
But when the multitudes saw it, they marvelled, and glorified God, which had given such power unto men." (KJV) Matthew 9:4-8
Notice "men" is plural. If the multitudes glorified God for giving this authority to men, only to later find out God did no such thing; but rather Yeshua was only referring to himself when he said son of man, "because he is also the God the Son which makes him the second person of the triune god", this would be a pitiful story.
The son of man is lord of the Sabbath
Another instance is when Yeshua and his disciples pluck seed on the Sabbath. They did not break Torah, but the Jewish leaders had made it a law that picking was considered harvesting. The Jews accuse Yeshua of breaking the Sabbath, and he replies:
"Have ye not read what David did, when he was an hungred, and they that were with him;
How he entered into the house of God, and did eat the shewbread, which was not lawful for him to eat, neither for them which were with him, but only for the priests?
Or have ye not read in the law, how that on the sabbath days the priests in the temple profane the sabbath, and are blameless?
But I say unto you, That in this place is one greater than the temple.
But if ye had known what this meaneth, I will have mercy, and not sacrifice, ye would not have condemned the guiltless.
For the son of man is Lord even of the sabbath day." (KJV) Matthew 12:3-8
Yeshua gives two specific reasons why he did no wrong. He appeals to David (a man) and those that were with him (men). He also appeals to the priests (men) and concludes "because the son of man is lord even of the Sabbath day".
You are the son of man
There is one instance where somebody besides Yeshua says 'the son of man' in the gospels. Yeshua is talking to some Greeks that had come to join the Passover, and he tells them that it is time for the son of man to be glorified. After a voice speaks from heaven, Yeshua says:
"And I, if I be lifted up from the earth, will draw all [men] unto me.
The people answered him, We have heard out of the law that Christ abideth for ever: and how sayest thou, the son of man must be lifted up? who is this son of man?" (KJV) John 12:34
They knew Yeshua was the Messiah, but they also thought 'the son of man' was a title that he used for himself. Yeshua didn't say 'the son of man' must be lifted up here, he said that 'he' must be lifted up. So they ask who the son of man is, and Yeshua gives them a very indirect answer:
"Then Jesus said unto them, Yet a little while is the light with you. Walk while ye have the light, lest darkness come upon you: for he that walketh in darkness knoweth not whither he goeth.
While ye have light, believe in the light, that ye may be the children of light. These things spake Jesus, and departed, and did hide himself from them." (KJV) John 12:35-36
Yeshua mentions only two things in his reply- 'the light' and 'you'. Yeshua cannot be the light, because nobody is ever called 'the son of Jesus'. The reason Yeshua says the light is in the world is because:
"Jesus cried and said, He that believeth on me, believeth not on me, but on him that sent me." (KJV) John 12:44
"For I have not spoken of myself; but the Father which sent me, he gave me a commandment, what I should say, and what I should speak." (KJV) John 12:49
For another example that the son of man is mankind, please see What did Jesus mean in Jn.9:37: "You have BOTH seen him and ..."?
Here are a few examples of how I've come to interpret these passages. They seem to confirm that this is the correct way we should understand 'the son of man'.
The son of man enters God's rest
"And a certain scribe came, and said unto him, Master, I will follow thee whithersoever thou goest.
And Jesus saith unto him, The foxes have holes, and the birds of the air have nests; but the son of man hath not where to lay his head." (KJV) Matthew 8:19
Either Yeshua is completely ignoring this man to talk about himself, or he means the man will get no rest if he follows him- because Yeshua doesn't have anywhere to rest. The second one is plausible, but it contradicts something else Yeshua says in Matthew; which I also believe to be the correct interpretation:
"Come unto me, all ye that labour and are heavy laden, and I will give you rest." (KJV) Matthew 11:28
The author of Hebrews also quotes the Scriptures to confirm this:
"For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world." (KJV) Hebrews 4:11 and Psalm 95:11
The son of man will rise on the third day
Another thing that Yeshua says a lot is:
"Then he took unto him the twelve, and said unto them, Behold, we go up to Jerusalem, and all things that are written by the prophets concerning the son of man shall be accomplished.
For he shall be delivered unto the Gentiles, and shall be mocked, and spitefully entreated, and spitted on:
And they shall scourge him, and put him to death: and the third day he shall rise again." (KJV) Luke 18:31
I have not found anything in the Scriptures that say the Messiah will rise again on the third day. There is only one verse that can possibly correlate with this:
"Come, and let us return unto YHVH: for he hath torn, and he will heal us; he hath smitten, and he will bind us up.
After two days will he revive us: in the third day he will raise us up, and we shall live in his sight." (KJV) Hosea 6:1-2
Son of God is a term used for both Yeshua and believers. It applies to those that believe in YHVH and obey Him. The son of man is all of mankind, including Yeshua. He uses this term in parables to teach us what mankind must go through before he can "be perfect even as He is perfect", and to show us what God has planned for us. Yeshua was the first man that God accomplished this in, and it's been His plan since the beginning:
"And God said, Let us make [accomplish] man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth." (KJV) Genesis 1:26