What is the difference between these two half bridge configurations?
The first circuit looks like a series resonant topology, probably trying to use the leakage inductance of the transformer as part of the resonant circuit.  Similar to an LLC, but hard to say exactly because we can't see the control scheme.

The advantages are higher efficiency due to soft switching, and the disadvantage is more complex control.

From the EETimes article here:



The second circuit is a conventional half-bridge.