It's hard to make something that is spinning around change its spinning direction. We say that the spinning thing has "angular momentum". Angular momentum only changes when the object is twisted ("torqued"), in the same way that linear momentum only changes when an object is pushed or pulled.
If the top isn't spinning, it starts with no angular momentum. Then the torque that gravity makes on it, if it's a little out of balance, just goes into making it turn sideways- in other words, fall over. If it's spinning, it's already got a lot of angular momentum along the spin direction. Most of the new angular momentum it gets from the torque of gravity just goes into making that spin point a little sideways, but no longer makes it fall.
Later, when friction has made the top slow down a lot, it doesn't have much angular momentum left. More of the torque due to gravity now goes to making the top flip sideways (fall) because just changing the direction it points doesn't change the spin angular momentum enough.
That probably sounds hard, and at first it is. Physics students often find that they have trouble grasping it without using some special math, but eventually it makes beautiful sense.
(republished on 07/13/06)