The products are completely separate. They work differently, have different account systems, they run on different operating systems, etc, etc. Tube Assist used to be called Tube Toolbox Web, but people kept getting it confused with Tube Toolbox (desktop) so we renamed it to TubeAssist.
If you have Windows and want more features and more control, then download Tube Toolbox and run it on your computer - http://www.tubetoolbox.com. If you're on a Mac, don't want to use your computer's resources or want to run the program on your phone/tablet, then use TubeAssist - https://www.tubeassist.com/.
If you have a Tube Toolbox subscription and want to switch to a TubeAssist subscription you will need to cancel your current Tube Toolbox subscription and then log in, create an account and create a new subscription at https://www.tubeassist.com/.
You are welcome to voice your opinion in the comments below which we will read and take into consideration.