The two most obvious effects of over-pitching are off-flavors. First, yeast material in excess quickly leads to autolysis, which has flavor by-products with very low flavor thresholds. In properly-pitched beers, this effect can take months to show itself. In beers with an excess of yeast solids, it can manifest in a matter of weeks - in fact, the time spent in a home-brewer's primary.
Second, tasters have observed thin beer, beer lacking in body and mouthfeel. To be perfectly honest, the exact cause of this effect is unknown, but it is strongly correlated with over-pitched yeast, so a connection is highly likely.
Third - and most important for the home-brewer - is suppression of esters. Yeast rely on the growth phase to reproduce enough cells to fully colonize the wort. In that phase, they use malt-based nutrients and the oxygen you provide during aeration to synthesize the components needed to build new cell walls during reproduction. While they're reproducing they're producing esters. All yeast produce esters, even lager yeast, and all beers benefit from ester production (yes, even lagers). Just because you can't taste as much ester from WLP840 as you can from Ringwood doesn't mean that WLP840 doesn't throw esters! Esters are absolutely necessary to beer, theory about "clean yeast" be damned.
When you over-pitch the colony doesn't need to reproduce. Thus measurably fewer esters are produced. This, while always detrimental to beer flavor, is noticeable in American and English strains and very pronounced with certain more flavorful strains, like Belgians.