I assume that this is one of the US English and UK English differences. Which is the most common phrase:
1) I need new "wheels" for my car.
2) I need new "tires" for my car.
Generally speaking, a wheel is a round object with a hub and an axle. A tire is the rubber part of a wheel that grips the road.
Not all wheels have tires. For example, a gyroscope wheel might not have a tire. An old covered wagon wheel would not have a tire, either, but the covered wagon would still have four wheels. A small lawn mower wheel might be cast entirely from plastic, so the mower may not have tires, either.
Wheels are for rolling (or sometimes spinning, consider a roulette wheel, for example); tires are for traction.
In an automobile, the wheels on a car consist of the rims and the tires. We generally replace tires, not wheels (the slang usage mentioned by WendiKidd notwithstanding). That said, a mechanic may explain how a transmission sends power from the engine to the wheels.
In short, the tire is part of some wheels.
As far as I'm aware there is no UK/US difference here, except in how we spell "tires"; in the US it is tires and in the UK it is tyres.
In normal conversation we would refer to them as tires, so your example sentence "I need new tires for my car" would be correct.
Wheels, on the other hand, can be used in a slang sense to describe an entire car. "I need a new set of wheels" doesn't mean you need new tires, it means you need a new car. So "I need new wheels for my car" doesn't make much sense (unless you're buying a car for your car). (This is at least true of AmE, anyway. This slang sense might not apply in the UK. But tires are tires (are tyres) no matter where you go!)
When you are driving along and you have a "blowout" or get a puncture/flat tyre, you need to either call your Roadside Assist Insurance provider or otherwise use your "spare wheel" (if you have one!!) to overcome the problem. Many cars these days do not even includde a "spare wheel"; not even an emergency wheel that is only rated for 80 km/h (~50 mph). Such cars only provide a Pressure-pack can that can (theoretically) inject a sealant into the tyre/tube and also re-inflate the tyre. If the actual tyre (the black, rubber thing that fits around the metal wheel rim), has been damaged, or even destroyed, by the sudden deflation of the pneumatic tyre, it will then be necessary to actually CHANGE THE TYRE. Changing the tyre involves the removal of the damaged tyre from the wheel RIM. The absolute minimum tools required to perform this operation, is a set of (two) tyre levers. It is also preferred that a rubber hammer/mallet. is also on hand. There must be videos out there that demonstrate how to actually remove a tyre from a rim, and how to re-install a tyre to a rim, so I will not try to explain it in text. The point is, it is a vast difference between changing a wheel and changing a tyre.