Why didn't young Barry disappear?

Little Barry didn't disappear because there was no reason for him to.

There is no "law of time" that states that only one person can exist in a timeline at any one time, because time simply doesn't work like that. Time doesn't care if two arbitrary combinations of matter floating through space-time make up what we as humans perceive as the same person.

When Barry travelled back to save his mom, since Barry is able to have very precise control over his location in the multiverse (for lack of a better of term), he was able to travel into his own past and see his younger self. If he had decided to save his mom, he would have made the specific decision to remain travelling on that timeline and travel to the future of that timeline. In the end, that's not what happened and he returned to his own timeline.

The real question should be: "What happened to future Barry?"

Incidentally, in "Out of Time" when Barry time travelled, it's not clear why the other Barry seemed to disappear. That's probably an inconsistency to aid in storytelling, although you could theorize that Barry was still travelling at that time and was "passing through", or maybe because Barry in that instance DID alter the past (by looking at himself), it created a timeline where the other Barry never went back in time, a type of paradox that the universe resolved by eliminating the other Barry. But more likely just an inconsistency.

we now know what happened to future BA as H.R. calls him he turned to a sad sack because Tracy brand did not build the speed force trap 4 years after iris's death and future Barry explained that even thought you will do everything in your power 2 save iris: you'll go and make time remnants but savitar will kill them, and there is no way to save iris because when he kills her it will feel like an eternity to him because she is the love of his life.