What player are you using for the MP3 part?

The original mp3 has been encoded with variable bit rate - change it to constant bit rate and try to sync it again. Constant bit rate mp3 keeps time much better than VBR, especially when jumping around to try to sync.

Yes that is correct

VLC isn't good at handling VBR MP3 files accurately. Convert the file to constant bitrate, then VLC works. I use VLCsync to keep the video player and audio player synced together.

For those struggling with MP3 sync this can be an issue with variable bitrate files. Many players (VLC in particular) can't seek properly to a point in time with VBR MP3 files. Either re-encode them to constant bitrate or just don't jump around.