Yes, the sect of 'the Way' is the same as the sect of 'the Nazarenes.'
Peter refers to Jesus as 'of Nazareth' in his sermon in Acts 2:
“Men of Israel, hear these words: Jesus of Nazareth, a Man attested by God to you by miracles, wonders, and signs which God did through Him in your midst, as you yourselves also know—
-Acts 2:22 (NKJV)
And there are quite a few other times where Jesus is associated specifically with Nazareth:
Then Peter said, “Silver and gold I do not have, but what I do have I give you: In the name of Jesus Christ of Nazareth, rise up and walk.” -Acts 3:6 (NKJV)
[Peter speaking:] let it be known to you all, and to all the people of Israel, that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead... -Acts 4:10 (NKJV)
[False witnesses speaking:] for we have heard him [Stephen] say that this Jesus of Nazareth will destroy this place and change the customs which Moses delivered to us. -Acts 8:14 (NKJV)
[Peter speaking to Cornelius:] how God anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him. -Acts 10:38 (NKJV)
[Paul speaking to King Agrippa:] Indeed, I myself thought I must do many things contrary to the name of Jesus of Nazareth. -Acts 26:9 (NKJV)
Acts 9 calls the followers of the Lord 'the Way':
1 Then Saul, still breathing threats and murder against the disciples of the Lord, went to the high priest 2 and asked letters from him to the synagogues of Damascus, so that if he found any who were of the Way, whether men or women, he might bring them bound to Jerusalem.
-Acts 9:1-2 (NKJV)
And Paul, recounting his own conversion on the road to Damascus, associates the 'Way' with Jesus of Nazareth:
4 I persecuted this Way to the death, binding and delivering into prisons both men and women, [...] 6 “Now it happened, as I journeyed and came near Damascus at about noon, suddenly a great light from heaven shone around me. 7 And I fell to the ground and heard a voice saying to me, ‘Saul, Saul, why are you persecuting Me?’ 8 So I answered, ‘Who are You, Lord?’ And He said to me, ‘I am Jesus of Nazareth, whom you are persecuting.’
-Acts 22:4, 6-8 (NKJV) {Note that Paul's name was 'Saul' at the time of his conversion}
So as you can see, followers of 'the Way' were followers of 'Jesus of Nazareth.' The sect of 'the Way' and the sect of 'the Nazarenes' are one and the same. Those accusing Paul in Acts 24 most likely tried to use this to their advantage, for even Nathanael, whom Jesus called a 'true Israelite indeed,' said in John 1:46 (NKJV): "Can anything good come out of Nazareth?"