I’m not the best person to differentiate between certain Southern accents (in the US), but I’m a bit confused by the difference between Bill Clinton and Mike Huckabee. They come from the same town in Arkansas, and yet they sound distinctly different to me. Is it that Huckabee’s accent is a bit more moderated towards standard American English and Clinton’s is just stronger? Or do they actually have different accents. If the latter, why?
A whole bunch of reasons, like:
What were the accents of the people around them while they were growing up?
Where they lived, and for long, other than Hope, Arkansas.
Whether they had courses in public speaking, diction, drama or anything else that might have affected how they speak.
My three children don’t even have identical accents and they grew up together in the same house.
When he was 7 Clinton moved to and was basically raised in Hot Springs, Arkansas. Except for K & 1st grade he attended all his pre-College schooling in Hot Springs.
To your point tho Hot Springs and Hope are less than 70 miles apart so I am not sure that is 100% what you are hearing … but it may be far enough apart for you to hear the distinction.
Early on Clinton also was raised almost entirely at times by his maternal Grandmother and her energy and drive probably left a big impact on him pushing him to read and excel etc. She was from Cotton Country and though Hope was "the big city " you could also be hearing Clinton with pieces of her accent.
Case in point:
I’ve lived in Michigan since I was a baby, but my parents are from New York. Even today (I’m 32) people will occasionally ask me if I’m from New York.
My parents have mostly lost their accents, but the old East Coast squawk will resurface from time to time. When mom’s had a drink or two, she can do a damn good Bugs Bunny impersonation 
I am from Northern Louisiana and have passed through Hope, AR and spent days in Hot Springs, AR. Hope is a very Southern and poor town in southern Arkansas yet Hot Springs is a minor city has a rather affluent past from days of yore with big resort hotels and the like. The have strong Southern accents there too.
I don’t think there is an explanation other than successful people build their own accents to suit themselves based roughly on their background and family. Bill Clinton’s accent is invented as is George W Bush’s. Having a Southern accent is a strong accent in American politics but neither of them have accents true to where they grew up. I grew up with a strong Southern drawl derived from the tiny Louisiana town that I grew up in but I lost it soon after I went to college and it was unconscious and not desired. My aunt has never spent more than a few days at a time outside of the same tiny Southern town in grew up in yet she has an accent that mixes Southern with basic Northern with Manhattan Jewish. No one knows why.
I live in Arkansas and spoke with Clinton. I don’t notice any difference in their accent. I’ll have to pay more attention to Huckabee, though. I spent a lot of time ignoring him. 
Let me clarify… I’m not sure if I’m actually detecting a different accent, or just a heavier and lighter version of the same accent.
Although I lost my accent years ago, I used to have a “light” New England accent. I would drop a lot of my Rs, for example, but you’d never hear me say Chiner (for China). I’m wondering if that’s what’s going on, rather than them having different accents. It’s hard for me to tell, because my ears aren’t too attuned to Southern accents.
I doubt that very seriously. It’s very difficult to make up an accent and not slip up regularly. I know it’s common for people to think Bush’s accent is fake, but the guy spent most of his life in Texas. He really is a Texan.
They may have made a conscious decision to try and not lose their accents, but I see no reason to think they’re not real.
I don’t mean that they sit down with a piece of paper and a speech coach and invent their own personalized accent for PR measures. I think it is much more subtle and unconscious than that. Bill Clinton and George W Bush have stronger accents than they should based on their academic backgrounds and dealings with people from all kinds of geographic backgrounds for many years. Accents tend to fade quickly under those conditions as you yourself acknowledge and I experienced myself. However, if a Southern accent was important to my life and job, I would probably have subconsciously found a way to preserve it.
Think about impressionists. All presidents in recent memory seem to be easy to impersonate down to the exact accent. I don’t think most people are that way such that a 1 minute impression would let virtually everyone know who is being impersonated. Everything about them needs to stand out in that way.
Arkansas has three geographical regions; the delta, plains and mountains…but Mrs. Plant just pointed out to me that is crap. Bill Clinton is a Rhodes Scholar. Huckabee has a Theological Degree from a second rate Arkansas college.
MY MInSc being from the esteemed Graduate Institute of Technology from the University of Arkansas, of course.
Let me give another example - the Kennedys. They have their own family accent and I have never heard a good explanation of where that specific accent came from. It just stays in the family and I have never met another person in Massachusetts that has it although I heard a woman with a roughly similar one once. Yet, Ted Kennedy maintains it just fine even after spending decades mostly in DC. JFK did too. That is a political thing to be sure although I won’t claim it is completely conscience either.
The Kennedys spent as much time in New York City as in Massachusetts.
That really showed in JFK Jr. He had a NYC accent, not a MA accent.