Well...I like the false bottom because it's definitely lighter than gravel (especially if you ever have to move the viv!) and since there's nothing under there, you have a lot more room for drainage than if it was filled with gravel.
If you have to break down the tank, you aren't stuck with a bunch of dirty gravel to dispose of.. false bottoms can be disinfected by scrubbing with a soft brush and then rinsing followed by soaking with a bleach solution...
They weigh much much less than a tank full of gravel... they are easier to remove surplus water..
I use all 3 methods (egg crate style false bottom, European sloping false bottom, and leca drainage layer) in various tanks.
Here in the UK the leca drainage layer seems to be the most popular, mainly because it's the easiest to do problably. However I personally prefer either a European one (as it comes ready installed in European style vivs. lol) or an egg crate style false bottom. You can do more with the egg crate style false bottom as it provides a space in which to house a water pump to drive water features, or to install a small aquarium heater and also seems to help to maintain humidity better. The down side is the work constructing them and making sure frogs can't get into them.
European false bottoms are just a sloping base with a drainage gully at the front (usually drilled so you can fit a drain set). Not really something that you usually add to a vivarium, more something that either came already installed when you bought it, or you put it in when building your own viv. The plus with them is they come ready installed, and don't use a lot of the vivs hight, and often the space beneath them gets used to mount a heat mat. The negatives, well they are pretty permanentand need to be added when the viv is constructed.
I only used the leca/gravel drainage layer in 1 viv to be honest, and didn't like it. Pros are it's very easy and fairly cheap to do, they also make it possible to easily create a small pool by digging it out a bit. Cons are it adds more weight to the viv and isn't as easy to install a drain in. They also take up a fair bit of hight to actually be useful, as 2 inches of leca wont hold as much water as 2 inches of empty space simply because the leca/gravel takes up space itself. Too often I've seen vivs with new folks trying this method where they just haven't used enough leca for it to actually do much.
So the winners for me are egg crate type false bottoms for tank conversions, or to just buy European style vivs with the sloping false bottom and drainage gully. It's a matter of personal choice though, lots and lots and lots of UK hobbyists swear by hydro leca layers.