Well there's a few things.
First, higher temperatures mean more volatile aroma and flavour compounds will be released, leaving a flatter-tasting stock.
Second, boiling means more motion within the liquid, which makes it harder to skim off the protein scum which forms on the surface while a stock is cooking. In addition, some of the scum will just become reincorporated into the stock via a process similar to emulsion.
Third, boiling induces faster breakdown of proteins and other particulate matter, leaving you with a cloudier end product. This is of course of very little concern for stocks which will end up incorporated into (opaque) soups or sauces.
Of the three, only the first is of any real concern to home cooks. The other two issues aren't a huge deal at the restaurant level (modulo need for clarity in consomme etc), particularly if the restaurant uses Superbags or algae filtration for clarifying stocks.