Is the wiring in this subpanel correct?
If you're in the US, this is NOT correct.
Enclosure not bonded
The bonding strap on the right side of the panel (when looking at the panel), should be moved to the left side. It should be connected to the enclosure using the green screw, and the tab should be connected to one of the grounding bar's terminals.

Double tapped neutral bar
On the neutral bar, only a single conductor can terminate per screw terminal. You've got two wires per terminal, which is not allowed.

Neutral and grounding conductors not separated
In a secondary panel, the grounding conductors and &quot;neutral&quot; conductors must be kept separate.
You should move the grounding conductors, and terminate them at the grounding bar instead.

Neutral and grounding bars not separated
You should also remove the bonding conductor that is connecting the grounding and neutral bars together.  (Thanks @ArchonOSX).

Unprotected wiring
I'm not sure what's going on at the top of the enclosure, but it looks like you've got some conduit work to finish up.


NOTES:

The main panel photo is fuzzy, so I can't tell what size breaker is feeding the second panel. You also didn't mention the size of the conductors between the panels. So I can't say for sure if you used the proper size conductors.
It also looks like you connected the ground to the neutral bar in the main panel, and the neutral to the ground bar.  Not such a big deal, since they're bonded in the main panel.
This answer is based on National Electrical Code 2014.