There are some words that are different, but I don't think you need to worry about it too much (lift/elevator, lorry/truck, vacation/holiday etc.) -- they'll understand you whichever word you choose to use.

(I'm a US native). The differences are small. A US native can listen to a BBC news broadcast and understand it effortlessly, despite a definite "British accent."

There are also difference between Massachusetts and Mississippi English in the United States, and London and Liverpool English in the UK. The reason why we distinguish between "British" and "US English" is mostly an ocean, national pride, and tradition.

The United States became independent of England in 1776. It was a literate country with its own publishers. A textbook author named Noah Webster had a lot of success with a spelling textbook, and decided there was a market for a dictionary published in the US. He compiled "An American Dictionary of the English Language." He decided to change and simplify a few spelling here and there: "colour" become "color," "centre" became "center," "traveller" became "traveler." It was a big success. The fact that we use different dictionaries has tended to create an exaggerated idea of the difference in language.

I have no trouble understanding British English unless is very local, full of slang, or spoken very rapidly. 

An acquaintance of mine from Guatemala says she had trouble understanding her Dominican neighbor at first.

I think the differences between US and British English are probably smaller than the difference between, say, Mexican and Argentinian Spanish. The differences in grammar are certainly no greater than the difference between "tuteo" and "voseo."