No matter whether you train with a "regular" Certified Flight Instructor or a Light Sport Certified Flight Instructor, you will be trained to the same standards in preparation for your Light Sport pilot practical test, otherwise known as your "checkride". There will be no difference in the restrictions that exist once you get your rating; they will be driven by the laws surrounding flying Light Sport. However, there is currently a practical difference between how the FAA views the training given between the CFI and CFI-S. Light Sport regulations were written "after the fact" of the establishment of other pilot ratings; as a result, there are often holes and disconnects between Part 61 Subpart rules and Part 61 and 91 rules. This is one of those cases. Since Subpart K allows CFI-S to train only sport pilots, the FAA is reading the regulations literally to say that no dual training given by a CFI-S counts toward dual training requirements for Private Pilot, even though the skills transfer and the Sport Pilot's total hours and Pilot In Command time do (and even though he got those skills because of his work with a CFI-S). The FAA is aware of the disconnect that has been protested by the major pilot organizations. However, as of this date, and because of the agency's inaction, this dilemna exists, much to the detriment of the folks who..like me..have labored to get their Light Sport Certified Flight Instructor rating and have passed the exams.
Yes, this ignores the fact that the manuevers for the Light Sport PTS were essentially taken from the Private Pilot PTS, that the aerial skills are exactly equivalent, and that some LSA's may actually perform better than most aircraft used for Private Pilot training. (In fact, some standard category aircraft that now fall into the LSA class were often used and are used as Private Pilot trainers.) It also ignores the double standard that already exists, i.e., as a CFI-S my only restriction on giving a BFR to a Private Pilot (or better) is that it is flown in a LSA under Light Sport rules--the pilot can take the BFR and go fly any airplane he is qualified in after that. The same holds true if I give such a pilot a tailwheel endorsement. In these cases, the FAA seems to consider the training as equivalent but not when training a Light Sport pilot for his rating. That doesn't make sense.
While the CFI-S dual hours currently do not count, stop and look at the situation realistically before making decision simply based on whether the instructor is a CFI or CFI-S. Frankly, though we all now that money talks, where this most comes into play is if your intent is to start in Light Sport and then progress upward. First, no matter who you fly with; that is always an option. Yes, the amount of money you spend may be different. Keep in mind, though, that anytime you move up in ratings, you're going to have to pay for additional training, even if you fly with the same CFI, which is probably the best deal you can get if that is your overriding concern. The total dual time required for the private is 20 hours. The new stuff for the private requires 3 hours of night flight training, 3 hours of insturment flight training, and the prep for the private pilot PTS requires 3 hours of time. Those are requirements that do not change regardless of whether your dual sport pilot time counts or not. (I am assuming here you are flying the same category an class of aircraft as you step up and not switching to a new one.) Total sport pilot dual hours required are 15 while private pilot requires 20, so you also are going to pay for the five additonal hours, also no matter who you fly with. So, whats the real difference? Hard to say, because it's also a factor of what your instructor decides he can "roll in" together, though I suspect that only might amount to a couple of hours. In any case, the salient point is...that despite what some folks will tell you...the real impact you have to consider from a "dual time lost" may be somewhat less than what's being portrayed. If someone starts the discussion talking about 20 hours of dual lost, be skeptical; this is using the Private dual requirements to perform a slight of hand. (You cannot logically argue this since the most that could be lost, even if you disregard everything else, is the 15 hours of dual you used to get your Light Sport rating.)
This is also not the only place in the pilot curriculum where you will see this kind of a "problem". If you go for your commercial rating, you will have to undertake additonal dual instruction and flights above and beyond what you got during your private pilot training. It's really no different, so keep that in mind when you are looking at all this.
This situation really shouldn't exist, but it does. It is based solely on a FAA letter of interpretation issued in 2009 that literally reversed the original intent of the rules...which was to let sport pilot time (including dual) count toward higher ratings. EAA, AOPA, and NAFI protested that interpretation and asked the FAA to fix it which they haven't done. We need every pilot we can get to keep GA alive; while the FAA guys will still have a job if it goes away, I don't think the airlines will ultimately do very well with only military pilots to feed their pipelines, esepcially considering how the military is pushing drones. We'll have to see how the public feels about getting into an airliner without a pilot or one who only flies them there remotely.
But I digress...:-)
Regardless, the best approach is to pick an instructor who knows his subject and you can work with the best, regardless of his rating. My experience has been that CFI-S's who fly Light Sport a lot understand their airplanes and the eccentricities of the Light Sport regulations better than most CFI's. (I've been to Light Sport presentations at major airshows, all given by "regular" CFI's, which had serious flaws and misrepresentations). That's not always going to be the case; it's always going to come down to the indivual instructor and how well you team up together. In the end, the alphabet soup isn't going to matter much.