With the mostly unsuccessful exceptions of the M551 Sheridan, AMX-13 Harpoon, and M60A2 Starship, most Western tanks did not use missiles. However, many Soviet tanks, from the T-55 to the T-80, used ATGM missiles that could in theory kill NATO tanks at a distance where they couldn't respond. Why would Western armies not emulate this concept?
Not sure if this counts as good answer mostly anecdotal from serving with armor, but NATO fire control systems were generally far superior to their Soviet counterparts.
There simply was not a need for a 'smart' munition when your main gun could accurately engage enemy tanks at range often while moving.
Also remember later on the use of TOW and Dragon missiles on other platforms (Bradley, M901 ITV, HMMWV, etc.) filled this role.