Both tournaments are open to anyone with a verifiable 190 average. The are many non-US nationals bowling in both events.
The USBC Nationals is an open tournament run by the USBC. It consists of 3 3-game events (5 person teams, doubles and singles). They also run a Master tournament the first week which is an individual only competition with qualifying rounds leading to a match format.
The US Open is a singles only event run by the PBA (and the Bowling Proprietors in the past). There will be a qualifying block (18 games I believe) to determine cashers and who advances in the tournament.
Conditions at the US Open are much tougher (flatter pattern) and you are moving pairs after every game.