Hmmm, I don't use those descriptions, but I imagine they are what I would call 'Technical Grade' (commercial) and Standard Laboratory Reagent, or SLR (Grade A).
'Technical grade' means that the acid has not been particularly purified or tested for purity. It would probably be good for a battery, but I doubt there is much difference in cost, at least for small quantities.
SLR is tested (and the results given) and has few enough impurities that they don't make any difference in most laboratory reactions.
If you want really pure stuff, for analytical work, there is 'Analytical Grade' which has more accurate testing, it is not always more pure. There is also a version purified (or at least tested) for use in the food industry (lead and other heavy metals often associated with the acid are lower concentrations). I have used some that cost around £200 per litre for trace element mass spectrometry, but that was special order stuff. Amazing being able to measure in picograms per litre though!
In short, garages probably use Technical Grade, and I bet their distilled water isn't all that pure anyway, so that is probably fine.
Small impurities can make a difference though, so if the SLR grade is only a little dearer, I would use that.
Just one other thing, if anyone offers you 'fuming sulphuric acid', don't use it. It contains free SO3 and is far too nasty!
Cheers,
FoxyRick.