A true counter can switch into the threat and consistently come out on top. 1 vs 1, they win the fight.
A check is a bit looser. It can come in and threaten the threat (be it by tanking a hit on the switch and outspeeding ftw, or tanking 2 hits and KOing, or tanking X hits and YKOing. A revenge killer can also "check" something.) The difference here is that the check will win 1 vs 1, but either A) can't necessarily switch in and win, or B) will be left severely weakened by doing so.
Both checks and counters force the threat out. However, a counter will be able to do so consistently throughout the battle, while a check will come out on top, but can't be counted on for the long haul. A threat can slowly wear down and break it's own checks. It ain't getting past a counter.