Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?
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closed as off topic by Hans Lundmark, tomasz, rschwieb, Emily, Aang Sep 23 '12 at 17:29
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The form of the question is somewhat iffy, but may be intended to ask a question that has a sense that I understand: since most representations are on complex vector spaces, a representation of a Lie algebra is indistinguishable from a representation of its complexification. The complexifications of su(2) and sl(2,R) are both sl(2,C)... often unhelpfully written merely as sl(2), not acknowledging the real-versus-complex game.
Perhaps another iteration of the question would make clearer what other/remaining issues might be...
Edit: The Lie groups $SU(2)$ and $SL_2(\mathbb R)$ are easier to distinguish: the former is compact and the latter is not, which has as consequence that the irreducible unitary repns of the former are all finite-dimensional (indexed by highest weight), while the irreducible unitaries of $SL_2(\mathbb R)$ are all infinite-dimensional (apart from the trivial repn). Appreciating this point about the Lie groups is probably necessary in order to appreciate distinctions about the Lie algebras which otherwise might seem capricious.
I can try to answer. Elements of $SL(n)$ preserve only volume, but elements of $SU(n)$ also preserve distance.