Why use embryonic neurons to study protein knockouts/mutants in long term potentiation?
Some suggestions, there may be more:

1) The knockout may not be viable to adulthood (the animals die). Perhaps heterozygotes are viable, but to test the full knockout you need a homozygote.

2) Even if the knockout is viable to adulthood, the brain may develop adaptations to the knockout that aren't specific to the knockout itself: up- or down-regulation of certain channels, for example. 

3) Plasticity is upregulated in developing brains compared to adult brains, so the effects may be more pronounced in the young tissue.

4) Younger tissue cultures better - you can keep cells alive longer, more easily.

Also good to note is that undifferentiated stem cells tend to be "neuron-default" - that is, if you just culture embryonic stem cells and you don't do anything special to them, they will tend to generate neuron-like cells. In the adult, neurons are always going to be alongside other types of cells: glia, vascular tissue, etc.