I've recently started playing again after 2 months off (due to tendonitis in my arm).
I bought one of those braces because I could barely hold my racket and I needed something to help me get through a USTA match I had (I didn't want to forfeit). Looking back, I should've saved the 20 bucks and just rested.
It did help with the pain during my match but I knew in the back of my mind that I was still doing damage to my tendons by playing through the pain. The problem with that brace is that it gives you the impression that you are "cured" when in fact it is just masking the pain which is your body's way of saying something is wrong.
Since then I've switched to softer strings (VS gut and Maxim Touch) strung at a lower tension as well as not using a death grip when I play. I feel a lot better now.
Most people use that brace because they want a "quick fix" (as I did) for their tennis elbow pain. They love playing and they don't want to take the time off to properly rest and fix the problem correctly. Take it from me, just rest up, ice, take some NSAIDs, and when you return, check to make sure your technique is correct. You don't want to rely on wearing stuff on your arm just so you can keep playing the sport you love most.