I don't know in game but in real life, wheels do affect performance in the following ways:
1. Bigger wheels tend to be heavier hence increase unsprung weight which reduce acceleration and deceleration performance a bit.
2. If the wheel is wider running on wider tires such as 275 series, this definitely increases traction hence better handling in general.
3. Bigger diameter does affect speedometer reading because bigger diameter will have to rotate a bit less to cover the same distance in a smaller wheel.
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I would say there is a bit more to it than that, for example
1. Bigger wheels are heavier. This is generally true, however a bigger wheel may actually make your corner weight go down... if it is a nice wheel that is light for it's size, even if it's heavier. Why? Tires are also quite heavy, and if you change to a tire that is the same width but shorter sidewall (to keep the overall diameter the same), your corner weight can go down even with a larger wheel. When you make the wheels wider it becomes harder to keep weight down. Many tire manufacturers will actually list the weight of the tire in the specifications (which you can find online) This way you can actually get an estimate of what your corner weights (unsprung) will be and if they will be going up or down.
2. Wider tires can increase grip, but in some cars (like very light weight cars) you can actually reduce the pressure on the contact patch enough to cause the tires to heat up to optimum temp too slowly or not at all. (in an extreme case) When choosing section width take into account the weight of your car. If you take it to autox or to the track, take tire temperatures after you run(also necessary for proper alignment adjustments) If they are very low you may benefit from a narrower wheel and tire (less unsprung weight) Wider tires also increase rolling resistance and steering effort. Keep this in mind if your car is also for the street.
[Another thing about grip.. the more of it you have, the softer your springs 'feel' in the corners. If you have stock suspension and throw on race tires the car will be faster but have more body roll. Make sure you plan for this. This is one of the reasons why it's becoming more difficult for OEMs to make cars that are comfortable on the street and easy to drive on modern tires.. finding the balance between stiffness is more and more difficult the more grip you have. IMO it's also probably one of the reasons the FR-S twins have Prius tires.]
3. Bigger tire diameter = taller gearing. This is true, this is why when shopping for wheels/tires you should usually try to keep the overall diameter the same, there are calculators you can find online just for this purpose. (for example my 195/50/15s on my AE86 are about the same diameter as the 185/60/14s that came on it. something like 23"? I forget, it's been a while)
I would also add that sidewall sizes will contribute a lot to the 'feel' of the steering. Shorter, stiffer sidewalls will feel 'sharper' most of the time, but will also be harder to control at the limit. (This is not universally true, as there are lots of different tires out there, but generally...) Shorter sidewalls also ride rougher and may even put your wheels in danger on rough roads. Try to find a good balance.
