An analogy may help. Imagine a rock perched at the top of a hill. If given a small push, the rock will roll all the way down the hill without any further input of energy (in fact, releasing energy as it goes). That's what we mean by a "spontaneous" reaction. The reverse (pushing the rock up the hill) is a "nonspontaneous" reaction into which you have to continuously put energy for it to proceed.
But now let's consider that small initial push. Just how small is it? If the rock is small and perched on a very sharp hilltop, maybe a strong breeze or a squirrel jumping on it is enough. But if the rock is big and perched on a rather broad hilltop, it may need an earthquake or big tree falling on it to get moving. Of course, earthquakes and big trees falling are much rarer events than a squirrel jumping around or high wind. So if the rock needs a tiny push, we expect it to come flying down the hill quite soon after it's put up there, and if it needs a big push, it will be quite a long time before it gets rolling.
This is what we mean by the kinetics of the reaction -- how fast it goes. A fast reaction needs hardly any push to get started, a slow reaction needs a big push, and small pushes are much more common than big pushes. (The "push" for a real chemical reaction is some fluctuation in energy in the environment, like the coincidence of a bunch of simultaneous high-speed collisions of molecules.)
Hopefully the analogy makes clear that the thermodynamics of a reaction (whether it is spontaneous or not) is completely separate from the kinetics (whether it is fast or not).
Finally, use caution when everyday words (like "spontaneous") are used in scientific language. The implications may not be the same! In everyday speech "spontaneous" often has the additional implication of "quick." If I said "I spontaneously break into song during meetings" you would find it odd if I only did it every 25 years. There's an implication, in everyday speech, of speed as well as no need for a driving force. But that implication does not carry over to the scientific use of the word!