I am currently working my way through probability theory, and stumbled upon a number of instances in which Borel sets/algebras/measures were specified (usually with Lebesgue integrals mentioned). So far I could not find out what exactly is to be gained from specifying you are working on a Borel set. My understanding is probably flawed, so I'd just write down what I've understood so far, and hope you could point out where I went wrong, or what I am missing. Here's what I have gathered so far:
$$\{ \mbox{Borel sets} \} \subseteq\{ \mbox{Borel algebra} \} \subseteq \{ \mbox{Sigma algebra} \} $$
Borel sets seem to be a special kind of sigma algebra, created from every single combination you can get from unions, intersections and complements on closed and open intervals. Borel algebras are a collection of borel sets, on which Borel measures are defined - I assume the need to make the distinction between sets and algebra arises from the fact that you can choose subsets of the Borel algebra and work with these. Apparently Borel sets are not power sets, although I still fail to see why.
Now the main point: Particularly in probability theory, what is to be gained from specifying that you are working on a Borel algebra over specifying you are in a sigma algebra? What is the information carried by this distinction?