Here's the advantages and disadvantages of each.

False bottom-
advantages - light weight making viv movement easier and less risk of cracking glass bottom, easy to shape, easy access to filters waterfalls etc, very inexpensive especially for larger builds
disadvantages - no surface area for "good" bacteria to grow to keep water relatively clean. (If you don't have moving water, it wouldn't matter)

LECA
advantage - pretty much wash pour and forget it. Looks better thru glass than egg crate. Provides area for bacteria to grow disadvantage - heavy heavy heavy. Doesn't matter if you don't plan on moving the tank. The weight could crack a larger tank.

Both methods are very effective for drainage layers. You have to take into account the size of your build and whether the tank needs to be moved around. I hope this helps.

I personally went with egg crate false bottom in the center with 1" wide layer of LECA all around the perimeter to hide the false bottom for my 40 gallon build. I don't think I have pics of that phase of my build. If I find any I'll post them for you.