Bud is acknowledged as the most influential post-war jazz pianist. He was the principle jazz pianist to influence Bill Evans, who in turn heavily influenced Herbie Hancock, Chick Corea, Keith Jarrett and many others.
Monk was Bud's close friend and mentor. As you know, however, Bud approached jazz piano much differently than Monk, displaying technical virtuosity along with the great composer's skill he shared with Monk -- probably partly a result of Bud's earlier training in European classical piano technique.
Try to find early recordings, early 50s/late 40s. Bud's health deteriorated steadily after he received a severe police beating in NYC in the late 40s. Although he continued to enjoy periods of brilliance almost until his death, they became increasingly rare over time.
He did a lot of work while living in France, and much of it is available on French labels, but the early stuff is better.
There also are a few solo sides (available on Verve?) which are sensational!
Bud Lives!