Instead of a board certified PS for BBL, I would like an experienced board certified general surgeon. What's the difference?

While there are many practitioners of cosmetic surgery, only a select few are certified by the American Board of Plastic Surgeons. The American Board of Plastic Surgeons (ABPS) is the only board certification in plastic and reconstructive surgery which is recognized by the American Board of Medical Examiners (ABME)
Certification assures graduation from an accredited medical school and completion of at least six years of surgical training following medical school (with a minimum of three years plastic surgery residency). They have also passed comprehensive oral and written exams.
Additionally, patients should verify that surgery is being performed in accredited, state-licensed or Medicare-certified surgical facilities.
There is absolutely no substitute for certification. If you are less confident in the results of the board certified physicians you have seen in consultation the American Society of Plastic Surgeons is a useful resource for seeking well trained and board certified physicians in your area. No matter, how experienced a particular surgeon may be, venturing outside your scope of practice is a risk.
Always discuss your concerns with a plastic surgeon certified by the American Board of Plastic Surgeons. Certification brings certainty!

Board certification 
This is an excellent question and a very important one.
My answers tend to be lengthy at times this may be one of those longer answers.
No single physician or medical specialty has a monopoly on any one procedure or treatment. Legally a radiologist can do heart transplants. In the reality that is usually not the case though. 
Most major surgeries are done in operating rooms. To get access to an operating room in a hospital or surgery center the physician has to have privileges. Privileges are only granted to physicians with appropriate qualifications. Unless a physician owns their own hospital or surgery center they are required to have appropriate training and background.  There are many physicians who do own their own surgery center or do substantial surgeries in their offices.
Since the scope of practice cannot legally be limited there needs to be some mechanism to protect the public. These include state medical boards and other regulatory agencies as well as board certifications.
The great majority of medical board certifications are overseen by the American Board of medical specialties.
Their website is listed below.
That's an excellent place to start if you want to get some further background on board certification.
All the different medical specialty boards overseen by the American Board of medical specialties require some type of residency or fellowship training or proof of competencies. The great majority of practicing physicians in the United States belong to one of the many boards overseen by this agency.
Just as physicians cannot be legally limited to their scope of practice neither does the American Board of medical specialties have a monopoly on creating medical boards.
As such, there are medical boards that are not recognized by the American Board of medical specialties.
One such board is the American Board of cosmetic surgery. There is not technically a residency in cosmetic surgery(there are fellowships in cosmetic surgery done after a residency).  The great majority of training in cosmetic surgery is done in plastic surgery residency. Other specialties that also include some forms of cosmetic surgery during their training are ear nose and throat or otorhinolaryngology and dermatology.
You would think the majority of the members of the American Board of cosmetic surgery would be plastic surgeons. They are not. Very few plastic surgeons are members of the American Board of cosmetic surgery.   The majority of their members come from different backgrounds like general surgery OB/GYN dermatology internal medicine emergency medicine etc.etc. etc.
I am not going to go on and on about which board is a real bored and which board is a fake board. I will let the consumer figure that one out. I will say that both the American Society of plastic surgeons and the American Board plastic surgery which are the main political organizations that oversee plastic surgery have spent much time and effort trying to educate the public as to what certification means and why it is important. Some would argue more could and should be done in terms of educatingbthe public regarding board certification.
So why does it matter if a doctor is certified?   Being board-certified is no guarantee that a physician is going to be of the highest caliber. But it is a very rigorous process that is very difficult to achieve.  For starters it requires that you've completed residency or fellowship training. Anyone can practice medicine as long as they have finished medical school and completed one year of internship. That's generally five years of training. To put things in perspective most plastic surgeons have had 10-13 years of training.
One note on facial plastic surgery. There is a recognized board called the American Board of facial plastic surgery. Most of their members are trained in ear nose and throat or otorhinolaryngology and have completed a fellowship in facial plastic surgery.
The great majority of surgeries done on the face and in the head and neck were developed by plastic surgeons. Over the years otorhinolaryngologists and other specialties have evolved and incorporated many of those procedures into their own specialty.
The first Nobel Prize given to a surgeon was to Joseph Murray, a plastic surgeon who performed the first kidney transplant.  Today a plastic surgeon would most likely not be given privileges to perform kidney transplants in any major hospital in the United States.
Who is best qualified to do what procedure? I've tried to explain why board certification is important but that still leaves choices of which type of doctor.
Here are my criteria.
If a physician can offer all of the alternative procedures, can take care of the any and all complications, did not learn the procedure by experimenting on people after weekend courses and has privileges to do the procedure in a hospital they are qualified to do the procedure.
There are many competent dermatologists , otorhinolaryngologists etc.. who do liposuction. Few if any of them are trained to do tummy tucks, mini tummy tucks, brachioplasties(arm lifts) etc...
You may have heard the expression if you're a hammer the world looks like a nail. 
Unfortunately if someone only performs liposuction, and not the other procedures, they are probably less likely to recommend a tummy tuck or arm lift even if those may be the better procedures.  
you also mentioned hand certification. Hand surgery is a specialty on its own that has developed from backgrounds from both orthopedic surgery and plastic surgery. Most hand surgery in the United States is done by doctors who are either orthopedic surgeons or plastic surgeons. Many of those have done an additional fellowship training and hand surgery. There also general surgeons who have done fellowship training in hand surgery and can be certified in hand surgery as well.
As one of my favorite teachers used to say if you want to be a plastic surgeon you should go to plastic surgery school.
Cosmetic surgery in general pays significantly more than any insurance based procedure. Call it creed or just a desire to earn a good living, many physicians from different specialties want in on the action. 
As such, cosmetic surgery, which at one point was synonymous with plastic surgery, has been picked apart and gobbled up by many sometime talanted and sometimes not so talented physicians from different backgrounds.  
So, do your research look for board certification, understand the scope of practice and limitations of different specialties.  
My last recommendation is choose as surgeon who has a lot of experience, good reviews, excellent before and after pictures and who is a patient advocate looking out for your best interest. That may be easier said than done but that's what we should all be shooting for.
Another one of my professors used to say "don't look for bargains when buying parachutes or cosmetic surgery"
This is in the end a situation of "Caveat Emptor( buyer beware)"
I hope I was of some help.
Good luck,
Mats Hagstrom M.D