Personal call...
From a materials, construction, and performance standpoint compared to t-shirts? I say yes. Same for shorts.
For any activity that generates sweat, I've moved away from cotton. It stays wet too long, and becomes uncomfortable to the point that I get irritated. Does a synthetic fabric make me a better player? No. Does it keep my mind on the game by me not being irritated? Yes.
Just be aware that like any product SKU map there is good, better, and best. While the base models will provide base funtionality, the more you pay the more you get. The higher priced tops will use different fabric combinations/weights and will change the cuts of the clothing to become tennis specific....an example being how the shoulder area is sewn to prevent restriction, etc...Marketing terms for these technologies include:
Nike / Dri-Fit = Moisture Wicking performance
Adidas / ClimaCool, Clima365, and ClimaLite = Moisture Wicking Performance
http://www.shopadidas.ca/helpTechPage.jsp.vr
Adidas / ForMotion = Sport specific material combination/cuts of fabric
Nike does not have a technologly name for the cuts of their clothes, they just implement them into the higher priced pieces in the sport's collection. For instance, the Nike 10" Athlete Tennis short, uses a combination of Dri-Fit and spandex to provide its shape and funtionality...etc..
On a side note, these sport specific cuts do tend to affect the sizing, just something to be aware of.
Back to your situation...I'd suggest going for middle of the road stuff that won't break the bank, and not leave you having bought the base model and being disappointed. Tennis clothing comes out on a quarterly basis, you might look for the last quarter's models at a discount too, to try it out.
This all being said, there does seem to be a fashion component to tennis that turns a lot of people off (me included), and makes them think there is more form than function, and regarding some items this is true...You just have to look at each item on a case by case basis. That's what makes a site like this so cool...There is a thread on the "fit" of Nike's Spring line in this forum that is a great example.