Radioactivity and nuclear fission Can an unstable atom undergo both nuclear fission and radioactive decay?
Other than the fact that that fission has to be initiated is
 there a difference between the final stability of the atoms?
 A: Yes.  Uranium-235 for instance decays via $\alpha$ emission, and is also, famously, fissile.  The results of fission and decay are not the same but, by definition, you end up, at the end of a possibly-long chain, with stable nuclei both cases (but not the same ones)
You could very easily have looked this up (the Wikipedia entry for 'Uranium' would be a good starting point).
A: There is a decay process called Spontaneous Fission.
U-235 for example, will undergo alpha decay about 99.99999999% of the time. But about 10^-9% of the time it will undergo spontaneous fission instead, where it breaks apart into smaller atoms. This happens without any outside forces causing it to do so.
So there is a difference between spontaneous fission and fission by other means.
Other fissile and unstable high mass atoms can also undergo spontaneous fission.
Through alpha decay the mass of the daughter product is likely still very high and will undergo alpha decay several more times. After fission the daughter products are both much smaller and therefore no longer would go through alpha decay. It is more likely theyd go through beta decay once or twice with relatively short half lives before reaching a stable state.
You can look up the U-235 decay chain. Through alpha decay it must go through a minimum of 11 decays to reach stability. Many of those have very long half lives. But if it fissions and breaks into cs-137 with a relatively small half life, then the cs-137 decays to ba-137 which is stable.
A: There are several nuclides, most of them not naturally occurring, which will spontaneously (radioactively) undergo either alpha decay or spontaneous fission. In addition to $^{~235}$U mentioned  by @Bigjoemonger $^{232}$Th is a naturally-occurring nuclide which decays by both paths.
Most of the plutonium isotopes have a small probability of SF in addition to alpha. Probably the most famous SF nuclide is $^{252}$Cf which has a 3% SF rate and 97% alpha.
You can research others at The Table of Nuclides published by the Korean Atomic Energy Institute. There is another at Brookhaven National Laboratory or International Atomic Energy Agency. ( I prefer using the one at the Korean site because of the quick info window on the right side of the chart. I find its formatting easier to get the specific information I need for a particular nuclide.)
