The code to bustimes.org is "open source", that is anyone is allowed to look at the source code that runs the site (available at GitHub, for those who understand Python and Django coding). You can also fork the program and run it hosted locally, and modify it as you wish so long as you also make the derived program open source as well, per the licence terms that bustimes.org code was released under. (Notably, I struggle to find the derived open source code for timesbus, other than an unmodified dev branch that does not have any of the updated code)
The people behind timesbus have forked (copied and modified) bustimes, and are seemingly attempting to "improve" it, but the site appears to be marred with stability and data completion issues from my cursory research.