I believe spontaneous means it happens on its own. You don't need any outside influence to get the isotope to decay. This term is sometimes used in contrast to stimulated.
Random means one cannot know precisely when the next decay will happen, though one can predict the probability of such events occurring in some time interval.
A decay process can be both spontaneous and random, but only one of these two options touches on the statistical nature of decay.
There's a small bit in the introductory paragraph of the Wikipedia page on radioactive decay.