The analogy that Nadler and I like is that if (oo,1)-categories
are like vector spaces, then model categories are like vector spaces with a fixed
basis and homotopy categories are like vector spaces mod isomorphism - i.e., dimensions
of vector spaces. Which of the three would you rather work with?

In more details I'll take the low road and quote my paper with Francis and Nadler:

The theory of triangulated categories is inadequate
to handle many basic algebraic and geometric operations. Examples include
the absence of a good theory of gluing or of descent, of functor categories,
or of generators and relations. The essential problem is that
passing to homotopy categories discards essential information (in particular,
homotopy coherent structures, homotopy limits and homotopy colimits).

This information can be captured in many alternative ways, the most common
of which is the theory of model categories. Model structures keep weakly equivalent objects distinct but retain the extra
structure of resolutions which enables the formulation
of homotopy coherence. This extra structure can be very useful for calculations but makes
some functorial operations difficult. In particular, it can be hard to
construct certain derived functors because the given resolutions are inadequate.
There are also fundamental difficulties with the consideration of functor categories between model categories.

Anyway, in characteristic zero, the theories of dg categories, Aoo categories and stable oo,1 categories are all the same, and provide a "promised land" between the two extremes.
(The problems with localization of the Fukaya category are very serious, but are issues in geometry - the existence of instanton corrections - and not in the theory of A_oo categories). Perhaps the most useful result in this context that doesn't have analogs for dg or Aoo categories is Jacob Lurie's oo-Barr-Beck theorem, which has lots and lots of consequences - such as descent theory or the generation result that Ben mentioned. Another is having a good theory of tensor products of categories and internal hom of categories, neither of which works in the two extreme theories.