What is meant by process "performance"?
Speed. Smaller geometry processes are usually faster. Smaller gate lengths and parasitic capacitances give higher speeds. The smaller gate has a lower resistance and so can drive the next gate faster. Much of speed is about the balance of resistance and capacitance smaller geometry transistors gain twice from this. By speed I am referring to the propagation delay through a layer of logic which in turn leads to maximum clock speeds.

Lower power consumption as mentioned, capacitances and therefore charge transfer (currents) are lower.