At international level, though, the duo were respected team-mates at England and put their club differences aside to help their country.
In fact, the closest England have come to success in recent years was when a 21-year-old Neville was in their squad at Euro 1996, with Adams as his captain.
And, according to Neville, Adams was better than anyone within the Three Lions squad at motivating a team-mate.
Neville, who is now part of Roy Hodgson's coaching setup within the national team, had taken to Twitter to do a Q + A with fans.
Other questions he fielded included those about England's current squad and other Arsenal players past and present.
Despite his former rivalry with the Gunners, the 39-year-old was full of praise for those from north London with which he played for England. The best goalkeeper he played with, for instance, was David Seaman.
But Neville selected a Manchester United player when it came to choosing someone who should have won more caps for the Three Lions. Indeed, that man was Paul Scholes.