The 1993 legislation was passed in Congress after adding an NRA-backed amendment. The 1994 legislation was passed despite opposition from the NRA.
When lawmakers were hashing out the Brady Law's details before 1993, in 1991, the NRA supported an 'instant background check' system as an alternative to a proposed seven-day waiting period when Congress was considering the Brady Handgun Violence Prevention Act. At the time, the bill's advocates saw the NRA's proposal as an attempt to weaken or kill the Brady bill, because the technology for instant checks didn't exist then. On their website, NRA wrote that they opposed the original Brady bill because it was nothing but a waiting period with a mandate to the States to conduct background checks only on handgun buyers. The NRA floated the idea of and supported an instant background check by the FBI on all firearm buyers. The Brady Bill passed in Congress, despite NRA's opposition. However, Politico noted that to get the Brady Bill passed, backers had to work with an NRA-backed bipartisan amendment that weakened the bill's background check system. This means that Biden didn't exactly beat the NRA on that one but compromised with it. In 1999, NRA released an analysis of the Brady Handgun Violence Prevention Act, which they claim did not live up to its name. They said the law did not prevent handgun violence.
Further, the 1994 crime bill, which included a 10-year ban on assault rifles, was vehemently opposed by the NRA. They lobbied against the bill, and Rep. Richard Durbin claimed that NRA lobbyists who were determined to save their jobs, threatened members of Congress that a vote for the weapons ban would endanger their political lives. The New York Times reported that the National Rifle Association promised to punish lawmakers who backed the ban. Representative Jack Brooks, the pro-gun Texas Democrat who was chairman of the Judiciary Committee, led the effort to defeat the ban but was ultimately unsuccessful. Politico notes that the NRA's fingerprints were not on the 1994 assault weapons ban, leading to a victory for Biden over the NRA.