One main difference is that American Sign Language tends to be more verbose than British Sign Language. This means that American Sign Language speakers use more words to communicate their thoughts, while British Sign Language speakers use fewer words but with greater emphasis.
Another difference is that American Sign Language has a more theatrical quality to it, while British Sign Language is more direct and concise. This is due in part to the fact that ASL originated from French sign language, which has a more dramatic flair.
Despite these differences, both ASL and BSL are equally effective ways for deaf people to communicate with one another. In fact, many deaf people are able to understand both languages perfectly well. So if you're ever unsure which sign language someone is using, just ask!