IMHO, this is my take on them from experience, I haven't read up on the technical sides of them, so no doubt I'll get corrected a good few times.
It really depends on the gcard and the drivers, but usually:
OpenGL +'s
Higher frame rates that D3D
Can handle more surfaces better, and is just in general more steamline than d3d
D3D +'s
Usually more stable
Generally looks better (I believe it supports more software and hardware based technologies)
Basically, because it is MS linked, seems to work better on Windows.
In general, if you are experiencing crashes, first thing I would do is check your drivers, if that doesn't solve it, check your BIOS (Have you got turbo on, or fast read write turnaround for graphics cards etc.?)
And one major thing to have a look out for in any bios is video shadowing, TURN IT OFF! May be great for your MS Office editing, but for games it one of the 7 deadly sins.
Hope this helps a little
NeoCOR