The short answer is that they are very different, but become quite similar if you 1) stabilize, i.e invert smash product by $\mathbb{P}^1_k$ on the homotopy side and invert tensor product by the Tate motive of the motivic side and 2) pass to rational coefficients. This is the analogue of the similar result in topology : the rationalized stable homotopy category is equivalent to the derived category of $\mathbb{Q}$-vector spaces.

The precise comparison result if you do those two operations was announced by Morel in http://www.mathematik.uni-muenchen.de/~morel/Splittinggrassman.pdf and a proof was written down recently by Deglise and Cisinski in the preprint http://www.math.univ-paris13.fr/~deglise/docs/2009/DM.pdf paragraph 15.2

Even the stable, integral versions are quite different : one way to quantify this is to say that spectra in $SH_{\mathbb{A}^1}(k)$ represent generalized cohomology theories - oriented ones like motivic cohomology, algebraic K-theory, algebraic cobordism but also non-oriented like Balmer-Witt groups, Hermitian K-theory - while objects in $DM_{k}$ represent only "oriented cohomology theories with additive group law" (in the sense of Quillen) : see e.g the Déglise-Cisinski preprint above, paragraph 10.3

For the different between unstable versions, a good simple example is the case of curves of genus greater that 1 : their effective motives are non-trivial (weight one effective motivic cohomology detects Pic ) while their unstable homotopy type is in a sense completely disconnected. This is essentially the reason why unstable $\mathbb{A}^1$-homotopy seems most interesting for "nearly rational" varieties, see the papers of Asok and Morel.