Before I can talk about how the asthenosphere and oobleck are different, I first have to talk about how they're similar. Both substances do not behave like ordinary fluids: they either grow stronger or weaker when we stress them more.

The first difference is oobleck will strengthen when you deform it, which is why you can smash oobleck with a hammer. We call an oobleck-like fluid a shear-thickening fluid. The asthenosphere, although it moves slowly, actually becomes weaker with greater stress; it's a shear thinning fluid on long timescales.

The other big difference is the obvious one. The asthenosphere is much, much more viscous than oobleck. But . . .

Hypothetically speaking, what if we had super-oobleck which was immune to chemical changes? If we could shear it fast enough it would end up being even stronger than the asthenosphere!

Here is also a nice video on understanding the basic physics behind oobleck and some other colloidal materials:
oobleck revealed