Why doesn't hydrazine contain a double bond between two nitrogen atoms?
Very interesting question! Let's start with the nitrogens in hydrazine being approximately sp2-p hybridized, flat. The p orbitals on each nitrogen would overlap in a double bond configuration - but the "double bond" has four electrons, two from each nitrogen. If you consider a molecular orbital, the bonding AND the antibonding orbitals are filled, so the net bonding is zero. Actually worse than that, because the filled p orbitals on each nitrogen are aligned in the worst way - maximum repulsion.

The clever little nitrogens rehybridize to become pyramidal - perhaps not completely sp3, but not at all planar. This rearrangement hardly affects the N-H bonds, but allows the p orbitals to accept some s-character, which increases one lobe at the expense of the other. And of course, the prominent lobes find a way to point away from each other and reduce electrostatic repulsion of the electrons. You could say that the double bond character is reduced/eliminated, but the bonding loss is more than balanced by reducing the antibonding. 

A useful comparison can be made with ethylene: planar, sp2-p, because there are only 2 electrons in the double bond.