Difference between push pull and half bridge
A half-bridge is a specific style of push-pull stage where the two active elements are in series across the supply and they alternately conduct to supply current to the load. The name usually refers to a switching output stage rather than one with linear control or only low power output.

The term push-pull refers to any type of circuit where there are two active devices that each drive one half-cycle of the signal - one pushes and the other pulls.

The devices can be in series across the supply or both be between the supply and ground and use a transformer to combine the two halves.  They can be be used in linear mode for analog signals or just on/off for switching purposes.

The active devices can be any type of device such as FETs, electron tubes (valves), bipolar junction transistors etc.

Here are some examples:

Half-bridge:



Transformer coupled push-pull:



Complementary push-pull:



TTL output stage: