Black holes sort of make sense. Too much mass and too much gravity, so nothing leaves it. But how and why (and where and when) can white holes exist? How can any body not let things enter it while it keeps throwing things out? wouldn't it just simply evaporate? And if black holes also continuously emit radiation, what is the difference between black and white holes? Please correct me if my thinking is going in the wrong direction due to ignorance (or stupidity).
They don't exist (assuming appropriate energy conditions). They're just formal solution to Einstein's equations that you get by analytically continuing the metric for an eternal black hole into the infinite past. Even if eternal black holes existed (which we have no reason to believe is the case), you could never observe the existence of the corresponding white hole, since it's in the infinite past.