These two guitars are really quite different.
LP is set neck, dual humbucker. Strat is bolt-on neck, 3 single coil. The neck on the LP is tremendously chunky compared to the thin Strat neck.
I myself own an Epi LP but have played my friends Stratocasters quite often and here's what I've noticed:
Strat is easier to play because of the neck.
You can get a more biting, precise sound from the Strat.
LP will give you a fuller, more bell-like sound (check Garcia circa '68).
I don't think I could just pick one or the other... they are totally different.
I only wish my LP was a double cutaway so that I could play above the 15th fret with ease (I have pretty short fingers).
a whole new gameball