Flutie's first stint with the Patriots began as a "replacement player" during the 1987 strike. He wa acquired from the Chicago Bears for a 9th round pick (the draft was longer then), and was desperate to keep his pro career going. He played 17 games for the Patriots over the next three seasons before leaving for the CFL. At the time, the Patriots were awful, and fans couldn't understand why the team wouldn't name Flutie the starter. But the fact was he threw more INTs (14) than TDs (11) during his time with New England.
His CFL career was legendary, winning 3 Gray Cups and 6 "Most Outstanding Player" awards in Canada before crossing the border in 1998 to return to the NFL with the Buffalo Bills. 1998 was Flutie's best NFL season, as he was named to the Pro Bowl and won Comeback Player of the Year while throwing for 2711 yards and 20 TDs with just 11 INTs. Unfortunately, he had lost some of his speed, and his INTs creeped up again - over his last three years as an NFL starter, 2 with Buffalo and 1 with San Diego, Flutie's TD-INT ratio was a pedestrian 42-37.
Flutie has also been criticized for playing locker room politics, especially in Buffalo, where Rob Johnson couldn't stand him. It's hard to know who's telling the truth in these cases, but pro athletes are intensely competitive; every player knows that there's someone looking over their shoulder, hoping for a chance to replace them. Flutie's 20 year pro career spanning 3 leagues (USFL, NFL, CFL) is a testament to his tenacity.
References: Stats from Pro Football Reference, CFL information from the CFL's official website.