Actually the accents are the same or similar, at least in the tidewater regions. Look, for instance, to the fact that in the 1930's until the 1950's, it was common practice in Hollywood to hire accomplished British actors and actresses to play major parts involving the Southern accent.
The non-rhotic-"r" (the non-pronunciation of the r in certain words when the r is followed by another consonant) is common to the vast majority of English speakers up until the mid-19th Century. The dropped-r at the end of certain words is also a commonality.
Where it gets trickier is the further you move inland. When you move closer to the Appalachians, you get more of a mix, which is what you're liking thinking of when you think of the "southern accent". That accent is mixed with French accents in some areas, Germanic and Scottish accents in others, and in some limited circumstances, Irish.
But think about the "plantation accent" or the "high-Southern" accent. That in and of itself is closer to British English in both word pronunciation, phraseology, and word choice than any other accent.
For instance, here are a slew of words which are similar to 18th Century British English -- "I swanny" (derived from archaic British "I swear on you") , "y'all" (derived from archaic british "All ye", or "Ye all"), and "I done told you" (derived from archaic british "I didst tell to/of you" or whatever else, depending on the verb).
There are actually several enlightening books written on the subject, as well as numerous academic papers. Look to works by William Ferris and John Shelton Reed