I know a lot of doctors recommend ibuprofen over aspirin because ibuprofen is perceived to be a safer NSAID. I'm not sure if it's actually safer in adults or only in children though.
GI side effects in general and bleeding in particular are higher in analgesic dose level of aspirin. Aspirin is a very blunt instrument - it is both anti-inflammatory and anti-thrombotic - it destroys your platelet function even in low doses - this can be a feature/benefit (ie stroke and coronary artery disease or MI) - but for pure analgesia it usually just means increasing bleeding risk for no benefit.
I don't believe this is a meaningful risk for a healthy person just taking a few aspirin. Certainly you don't want to consume too much (like all medicines) nor use it chronically (unless the alternative risks are worse).
