There's the Duran ones, although they've appeared on DVD already, so I won't bother listing them.
Cutting Crew - I Just Died in Your Arms. There's one version in a warehouse, and the other version is "artsy".
Depeche Mode - People Are People, Different Mix (from Some Great Videos VHS/laserdisc)
Frankie Goes to Hollywood - Relax. One is in a gaybar, the other starts with FGTH neon lights + Holly Johnson dancing in front of lasers.
Hart, Corey - there's two versions of It Ain't Enough. The video and music are different on both; one is the album version, the other is the single remix.
Idol, Billy - Hot in the City. Supposedly, there are two versions ->
http://en.wikipedia.org/wiki/Hot_in_the_City
Mitsou - Bye Bye Mon Cowboy. Original Canadian version + US version (I think US is remixed audio).
Parachute Club, the - Rise Up. One is "live".
Platinum Blonde - Crying Over You. There's a live version of this also.
Pretenders, the - Don't Get Me Wrong. Both are similar, but the second version cuts to shots of the band.
Propaganda - Dr Mabuse - There's the b/w silent movie version, and then there's the sucky version that's in colour.
Talk Talk - It's My Life - Both are almost identical, but the second version shows snips of the band, whereas the original doesn't.