The bacterial spores you are referring to are called endospores. The spore is formed during unfavorable conditions, by duplication of the bacterial chromosomes, and consequent envelopment by various membranes and protective coatings (while still within the original cell). The spore is dormant and remains so as the outer cell degrades during harsh conditions, releasing the spore. I guess I'd call the spore a clone of the original?
On the other hand, cysts are formed through differentiation of the bacterial filaments into round structures that gain a cell wall. The cytoplasm is also changed in the dormant state. Cysts are less durable than endospores.
You may enjoy reading the relevant sections in this book, which go into greater detail.
edit: as Souvik has pointed out, this is only one type of "protective" spore. Actinomycetes also form spores, via differentiation of the cells, and cyanobacteria form akinetes also via differentiation (but this structure is not as durable as endospores).
Information about cyst formation is difficult to find, but in bacteria it appears to develop via changes in cell wall composition.