The continuity errors don't bother me, but one that REALLY stands out is Voyager's "The Q and the Grey" vs. TNG's "True Q".
In "The Q and the Grey", the Q have never reproduced. Our Q wants to be the first to do so. In the earlier episode "True Q", Amanda Rodgers learns that her parents were Q.
(*facepalm*)
That brings up another issue, where the Farpoint mission seems to be a first contact meeting between humanity and the Q. Yet the Q must have been messing around with humans a couple decades before when Amanda was born.
And VOY later showed that with the powers of the Q, they have been popping up all along in humanity's history.
And that makes me wonder why the Farpoint mission is when the Q decided to announce themselves and put humanity on trial. Why then? Not during the 21st century? Not during TOS? Not the 26th century?