I'd say generally MSR-H's surface is better-maintained than TWS, but even at TWS most of the problem areas aren't on the racing line. Neither is going to destroy your tires (as in, cause them to prematurely rip or tear or something). You'll destroy your tires the natural way at both tracks, by burning through the rubber in every corner
In general I'd say MSR-H, especially given it's closer to you, it's the better track of the two to learn on initially (because on average its runoff areas are more forgiving, although both have dangerous areas where you don't want to spin or run off at all). It also has better visibility; you can see about half the track at any given time if you lift your vision up and look around (which instructors should be helping train you to do!).
The fastest straight at MSR-H also isn't as fast as the main straight at TWS (especially Clockwise, but CCW is much more common at TWS), so TWS requires quite a bit more balls at the turnin at the end of the main straight, and can put more heat into your brake system.
But still, they're both good tracks to learn at! These are just finer points of distinction between the two.
Here's a video of me running my 370 recently at each one, both CCW:
MSR-H CCW:
https://www.youtube.com/watch?v=rV36RzGYOg4
TWS CCW:
https://www.youtube.com/watch?v=BdClfIBbFlc (this one's old but it's the last one I have good video for! I've dropped 2-3 seconds a lap since then).