TC serves four functions, at least in my opinion:
1) Stopping dry hits. Set your temp to 400-475 and you can vape your coil almost totally dry without burnt hits. Cotton begins searing around 420F (cotton burn test everyone on YT does), but you can go a bit higher than that before it gives a really nasty flavor. With TC you can vape a wick totally dry and not taste burnt cotton. This is why it is especially good for dripping.
2) Avoiding formaldehyde and other nasties that form when you overheat PG/VG. Remember any substance can thermally degrade if you get it hot enough. It is well known that VG can degrade into acrolein (nasty stuff), formaldehyde and other carbonyl compounds (none of them good to inhale). The exact temperature this happens in our atomizers is a sort of a guess, but John from Evolv (guy who invented TC in mods) said formaldehyde forms in our attys between 430-450F (depending on a few variables like PG/VG ratio, coil setup, etc). I don't know where he got this number, but he may have done internal testing or has connections to other scientists (he is working on developing the official reference e-cig for the National Institute of Health through a government grant. Even though he works for Evolv, I am inclined to listen to his input on this subject and I feel he provides legit information). In any case, whether his numbers are correct or not, we know for a FACT that formaldehyde and other nasties will form if you get PG and/or VG hot enough. For health conscious vapers, TC is a way for them to try and minimize this. Perhaps with future research we will know a lot more about safe vaping temperatures (I am positive scientists are studying this very issue right now). However, the best we can do now is throw out a ballpark figure.
3) Flavor adjustment. Some juices may taste better at warmer or lower temps.
4) Consistency of the pull. When you take a drag off a TC mod, the pull is 100% consistent in flavor, temp, and saturation no matter how long you pull on it. Without TC, the end of the hit typically gets dry (at least in most tanks) and the temperature of the vape fluctuates dramatically depending on your draw. It's not consistent. TC solves that.
Downsides to TC:
1) It may be cooler than you like. I was never a super sub-ohmer, but I don't find anything under 475F satisfying. Really I am more accustomed to 500F+. Right now I am rocking 450F and trying to accustom myself to it. I think I can get used to it without much issue.
2) Most TC mods (except the DNA-200) do not allow you to use the full power of the mod in TC mode. For instance, the mod might do 100 watts in Kanthal mode, but only 50 watts in temp mode. I think this stems from a misconception that watts = temperature. It doesn't. You can design a coil that will take 100w and never go above 450F if you really want to. You don't have to sacrifice vapor production for lower temps (though you will have to build big coils).
I suggest buying a cheap TC mod. Tons of them to be had for anywhere from $40-$75. If you like it, then you can step it up and buy a DNA-200.