There are many British dialects - older words are used less and less with the spread of standard education and the influence of mass communications, but accents tend to persist for longer. It's still often possible to tell the region someone comes from from their accents, and regional accents are still often heard on TV - especially, at present, the "Geordie" accent of the North East (Tyneside) region, which has become fashionable in advertising and announcing.
When people from overseas say "a British accent", they usually mean a general form of educated, non-regional English, which in this country tends to be called "Standard English", "BBC English", "the Queen's English" or "RP" (RP is a rather technical linguistic term which stands for "Received Pronunciation". (The second type of British accent which people may know, is the "Cockney" or London accent, often used in films. The only films where people from overseas are likely to hear traditional South- or West-country accents seem to be Pirate films, where, for some reason, the sailors are often shown talking that way...)
The traditional Sussex accent has largely disappeared, and been replaced by a general South Eastern, urbanized, accent, similar to London English. For example, the word "fern" used to be pronounced "varn"round here, but you won't hear that nowadays. My grandmother used several dialect words that you won't hear today - for example she called an ant (insect) "an emmet", and a newt (salamander) "an eft". My own accent is basically Standard English, but I do use some pronunciations and spellings that would be considered old-fashioned or regional.
If you're interested in the traditional Sussex dialect, there's a page on Wikipedia, explaining our county motto "We won't be druv" (driven) - which earlier would probably have been written "Us wunt be druv" (the form I was taught). On this page there's a poem from a hundred years ago, which illustrates some of the dialect.
(The line "Mus Wilfred come to Sussex, us heaved a stone at he" is a reference to Sussex being the last region in Anglo-Saxon England that was still heathen - St Wilfred, a Northerner, came to convert the people, but was considered arrogant and resisted - he was only accepted when he showed love instead of superiority)
We wunt be druv - Wikipedia, the free encyclopedia
Today, the main differences you may still hear in Sussex speech include the adverbs "somewhen" and "anywhen" (as in "I'll see you somewhen"); the use of "twitten" for a narrow passage between fences or buildings; and the use of "trug" for a traditional wooden basket. These were once used for carrying almost anything, especially by women, but are now mainly used for gardening or fruit picking.
Sussex trug - Wikipedia, the free encyclopedia