Snodin started his career as a trainee at Doncaster Rovers along with his brother Glynn under manager Billy Bremner, playing in midfield for the club. In September 1982, he scored a hat-trick in a 7-5 victory over Reading at Belle Vue, claiming the match ball ahead of Kerry Dixon who had scored four for the visitors. It would prove to be the only hat-trick of Snodin's career. Even though Doncaster were in the third division at the time, Snodin earned several call-ups to both the England Under-21s and the under-23s due to being such a strong player at such a young age. He soon caught the interest of the Leeds United manager Eddie Gray and was transferred to the club in the summer of 1985 for £200,000. Bremner soon followed Snodin to Leeds and appointed him as captain, replacing Leeds' legend Peter Lorimer who had retired from the game. Snodin added class and bite to United's midfield and became the key player in the side.
It wasn't long before Division One's clubs approached Leeds for his services. Everton and Liverpool both offered £840,000 in 1987, and cash-strapped Leeds accepted. Snodin chose Everton as his next team despite having agreed terms with Liverpool, and moved to Goodison Park in January 1987. Snodin helped his new club to win the title in his debut season. His never-say-die attitude was well received by the Everton fans, but it wasn't until Snodin played as an emergency right-back that he flourished for the club. This transformation to defender happened so quickly and with such ease that in February 1989 he was called up into the full England squad for a friendly international against Greece. Unfortunately Snodin was forced to withdraw due to injury and his problems worsened a few weeks later when he was carried off during a game against Sheffield Wednesday with a serious hamstring problem. Despite lengthy periods of rest and several operations, Snodin struggled to regain his fitness and spent the whole of the 1991–92 season on the sidelines.
In October 1994 he spent a while on loan with Sunderland and then in January 1995 he moved to Oldham Athletic, ending an association of nearly eight years and almost 200 first team appearances. He later played at Scarborough before returning to Doncaster Rovers in 1998 to manage his home town club. He managed the club for 18 months before moving into radio commentary.