The word nachash, Exodus 4:3, is the same word used to refer to the entity in the garden of Eden responsible for sin coming into the world. The word emphasises the hissing of a snake and the glistening scales of a snake which mesmerises its prey, fixating it before it strikes to poison its victim with venom.
This word is used in relation to Moses' rod, who grasped the tail of the serpent and it turned back into a rod which was then called 'the rod of God', Exodus 4:20.
The serpents sent among the children of Israel, Numbers 21:5-9, were also called nachash modified with the word saraph 'fiery'. The remedy to this plague of deadly serpents was the lifting of an 'ensign' on a pole, the likeness of a serpent in brass.
This figure is explained in the New Testament as being a figure of Christ, who takes away the sin of the world.
And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: That whosoever believeth in him should not perish, but have eternal life. [John 3:14,15 KJV]
The word tannin, 'serpent', Exodus 7:9, is used in singular form fourteen times in scripture and its concomitant plural or collective form tannim, appears, also, fourteen times giving a total of twenty eight times, seven times four, an indication of something that perfectly operates to all four points of the compass, north, south, east and west.
Various renderings are dragon, whale, sea monster and serpent.
The word tannoth is translated 'dragon' once in scripture and the word tannur is rendered as 'furnace' four times and 'fire' eleven times.
Thus the image is of something gigantic, something fiery (in itself, thus not the association of an adjective, saraph) and something associated with that which is tempestuous, fluid, moving, not solid.
... I hated Esau and laid his mountains and his heritage waste for the dragons (tannoth) of the wilderness. [Malachi 3:1 KJV]
I would see, here, similar concepts to those expressed in the New Testament Greek scriptures where the Entity referred to by different descriptions (but never named) is called both 'serpent' and 'drakon'.
Moses, representative (in this account) of a Deliverer, has power over the nachash (serpent). Aaron, representative of the true High Priest, yet to come, has power over the dragon.
Aaron's serpent swallows up the serpents of the magicians, showing that the Serpent over whom Aaron has power, by God's gift, is the head and the power of the lower entities over which the magicians have a more limited power, through sorcery.
Paul shook off a serpent which bit him, Acts 28:5, shaking it off into the fire, surely a figure of the faith that shakes off the attacks of the wicked one, whose end is in the lake of fire yet to be kindled by the unquenchable wrath of the Almighty.
Paul, in another manner, speaks of being delivered out of the mouth of the lion, 2 Timothy 4:17, another way of expressing his deliverance from the power of the enemy, even when the enemy has drawn near enough to close his teeth upon his prey.
So, I would see in the different words, applied to Moses' rod and Aaron's rod, figures which are carried forward into the New Testament and have considerable bearing regarding the ongoing war between the Entity, called Serpent, Satan, Drakon, Apollyon and Abaddon, now cast out of heaven by Michael and his angels, Revelation 12:7, and cast to the earth to do his worst, but only till Christ returns with clouds (of witnesses) to take the kingdom and to rule the nations with a rod of iron.
The serpent is in the hand of the man of God. In his hand, it is a rod - a rod of correction, a rod of chastening - and with this rod lies deliverance.
... with their hands, they will take up serpents ... [Mark 16:18 KJV]
All references to 'scripture' and 'renderings' are to the KJV.