I think GH's link and K.Buzzard summary of Labesse-Langlands explains that there is no easy comparison possible.

But I think at the heart of your question is something else. Are you asking about a decomposition into cuspidal, continuous and residual part? If yes, the decomposition is completly analogous (of course for technical convenience you should rather fix a central character - say trivial - in GL(2)).

You get

1) a direct sum of cuspidal representation (all single multiplicity)

2) a sum over the one-dimensional representations $\chi \circ \det$ with $\chi$ Hecke character and $\chi^2 =1$ resp. for SL(2) only the trivial rep.

3) a direct integral over parabolic induced representation $\chi_1,\chi_2$ with $\chi_j$
Hecke quasi character and $\chi_1 \chi_2 =1$

The proof in Gelbart-Jacquet "Analytic ..." translates easily to this situation.