Turbocharger or Supercharger
 

Which one is overall better?

And which one would be better for my 03 Mustang GT?

Everyone tells me turbo but im concerned about heat and maintenance, along with turbo lag.

I think it as whatever you like overall. Superchargers use up around 5 hp of the what you are producing to turn up the supercharger. A turbo however runs hot from the exhaust so you have to cool it somehow so if you think about it it adds weight to cool it and you have too spool up the turbo for power versus instant power from a supercharger.

So it's whatever you like.

Two entirely different animals. A turbocharger will deliver a great power punch once it spools up (goes to positive pressure,) and starts forcing air instead of sucking it into the engine. Plus a turbocharger doesn't hurt fuel economy.
A supercharger doesn't spool up, it's always on as it's driven by a belt, so the power punch is always available. No waiting, and that's absolutely critical. There is a significant fuel economy price to be paid with a supercharger but you don't get something for nothing in the car world. More of this means less of that.
As for modifying an 03, I think I'd just buy an '11 or later and get just as much power from the factory as going to all the expense and trouble of turbo or supercharger.

This biggest pro to a supercharger setup is compact layout and the aforementioned lack of under hood heat issues. The cost is in parasitic drag which can consume a lot of power when your talking big boost and lots of blower RPM. The most efficient blower design is a twin screw and they are also the most adiabatic efficient forced induction device out there.

On a roots or screw type supercharger the hit in fuel mileage can be worse compared to a centrifugal type since you creep up on boost earlier demanding more fuel but if you like a fat power curve they are great. I've messed around with cars using a centri with more total power and despite having about 150 less peak power I could keep up nicely until they started turn some serious RPM.

Dang Bob I had to look this word up! Adiabatic

If I had the money I would go with a turbo. You an make a lot more power