Firstly, there is no "Southern" accent, as the variation between a West Texas accent and an Augustan accent are pretty extreme. There's also Mississippi Delta, Blue and Smokey Mountain, Carolinian and, of course, Cajun, to name but a few. Some are only subtly different, while others are different enough to count as dialects.
Secondly, Hollywood's version of 'Southern' accent is terrible and usually completely lacking in the subtleties found within the real ones. They have gotten better at this over the years, but Tinseltown still has to get over its bias against real Southern accents.
Southern accents on the whole are softer on the R's, which is why some words seem to be pronounced similarly to English accents. It makes it easier for some British actors to pull it off. However, it's also quite divergent in pronunciation, cadence and tonality. The flat A's, soft consonant stops and heavy drawl, make it sound quite unlike any English accent.
It might be better to say that Southern accents are the least divergent, owing to the lack of fusion with immigrant accents that define the Northeast, but not that they're anything like any modern English accent.
The most English-like accents in North America tend to be concentrated in coastal New England and around Newfoundland, as well as some parts of Virginia.