See the link on the home page of this site. Terry has a great summary of the various types of pipes. To summarize his summary on copper pipes, type M is the thin wall and is used for inside plumbing. Type L is thicker walled and is used for outside water service, and type K is the thickest walled and is used for water mains and meters. Fittings are universal.
There would be nothing "wrong" about using a type L in the interior, but there is no advantage or benefit in doing so. Obviously as the thickness increases so does the price. Cutting requires more effort as the thickness increases, but tool used is the same, and I suspect it may take a bit more time to heat a joint to solder. Since the outside diameter of the three types remains the same, the inside diameters will decrease as the wall thickness increases, but it is such a small difference, you would not be able to tell by performance. Type M is more easily found, box stores always have it. To get the heavier weights you would probably have to go to a plumbing supply house.