What is their differences? In Britain English and American English, are their use the same?
In the United States there are really no rules. Street, road, avenue and more are used with no particular pattern. I can't think of any rule that doesn't have exceptions, and I think all of them are subject to fashions and regional variations and the whims of real estate developers.
In New York City there is a regular pattern: "avenues" are wide streets that run north and south, "streets" are narrower street that run east and west. This pattern is not universal. It is not even common.
The street I live on is a narrow, low-speed residential street in a not-so-fancy part of town and it is named as an "Avenue." Some other residential streets that come to mind in towns I've lived in include "North Street," "Birch Road," "Elm Drive," "Wyndham Lane," "Boniface Circle," "Oak Terrace," "Atwood Court," "Tuscany Trace."
People will tell you that a "Boulevard" is supposed to have a median strip but there are plenty of "Boulevards" that are just ordinary roads without so much as a white line down the center.
And don't let's get started on the differences--if any--between a "highway," a "thruway," a "freeway," a "parkway," a "tollway," a "turnpike," etc. except to say they are all high-speed, long-distance roads.
For the most part they are the same, but some words are spelled differently (E.g. color instead of colour, and mom instead of mum...). They speak in a different accent as well of course, and they use a few different words to us here in Britain (e.g. vacation instead of holiday). We understand each other just fine though because, as I said, it is mostly the same :).
I don't know if this is official, but usually roads are not within cities or towns. For example, a country road. I think most roadways in a town or city are streets, and the extra big ones can be named avenues or boulevards. There are highways, freeways and tollways too, but those are large roadways with multiple lanes, high speed limits and limited exits.
A road is what you drive along, either within a town or between towns. A street is more likely to have houses and shops on it. However, when it comes to naming conventions, such as the difference between Green Street, Green Road and Green Avenue, the use is fairly arbitrary. There is actually no way of knowing whether 'Green Street' is longer, shorter, wider or narrower than 'Green Road'.
The naming conventions are more or less the same in British and American English. The main difference I can think of is that 'Avenue' is more likely to be a wide city street in the US (think New York), whereas in the UK we don't tend to use 'Avenue' in this way. If your address is 'Green Avenue' in the UK, it's more likely to be a quiet residential street than a grand boulevard.