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Less unsprung mass = less vertical contact patch load variation (managing vertical load is arguably the whole entire objective of racing car design)
Allows a higher aspect ratio & bigger sidewall (fast road cars run low profile tyres for looks, but oh wait... F1, GP2, F3, Touring cars, GT cars etc all run average to high profile tyres!)
Cheaper tyres
Less overall weight
Reasons to want bigger wheels:
Bigger brakes
Vehicle handling will be upset less by bumps
Wider tyres
Opportunity for a wider track width
Small wheels generally win unless you can change the car so dramatically (e.g. you're a BTCC, WRC, GT team with hundreds of thousands of pounds to spend) that you can have all the benefits of bigger wheels, plus the space to fit high profile tyres (to get the usual benefits of a smaller wheel) and the money to buy proper motorsport wheels to offset the weight gain.
Thers nothing "wrong" with running 16's on a clio. 15's just offer more choice in tyres... and they are cheaper.
The weight is marginal, but its more of a ride issue on the road than a race/speed issue IMO. I have sets of both sizes and its not a simple answer of "15's is better". I run 15's on the road and 16's on track (opposite to most, probably) and I feel that its more an issue of the tyre width - you have a choice, 195 or 205 in both 15 and 16" wheels and a well set-up car will benefit on track from the 205 width (again, just my opinion). Choising the right tyre to work with your car is more important than wondering if you should be running 15 or 16" rims as the compound/constuction of the actual tyre is more improtant that the actual dimenssions when it comes to performance.
I personally think you are missing the point a little for a track/race car, whilst also making a valid point for road cars.
It's not going to save you 30 seconds a lap. There's no need for road users to go around burning 16" wheels & tyres. But given that the OP is building a race/track car from scratch why would you choose to buy wheels/tyres in a worse size for more money?