Well since Realms Beyond uses hexes I'm obviously supposed to argue in favor of hexes but personally, I think both systems can work equally well but it depends on implementation, of course. They both have their advantages and their disadvantages.
Squares obviously have the issue of dealing with diagonal movement. Does it cost the same amount of movement/action points as straight movement? Then in many cases walking the same distance diagonally will be cheaper in AP cost than walking it in straight lines. If you make it cost more than 1 AP, how much more will it cost? Diagonal movement on a square grid is more problematic to implement than on a hex grid.
Same goes for diagonal attacks. Do you require a reach weapon to attack diagonally, or is a normal melee weapon fine? What about the range of ranged weapons?
Square grids are also, well... square, which can force level design into very blocky shapes depending on whether all your assets confirm to a square grid, too. Look at games like Eschalon, the Spiderweb games, the early Ultimas and other similar games. There aren't any diagonals in these games, everything confirms to the square grid - no round shapes, no hexagonal or octagonal shapes, etc. This issue can easily be solved by allowing assets to be placed more freely and occupy half a square (which for the purpose of movement would count as occupied), so you could have, for example, diagonal wall tiles that occupy the halves of two squares rather than one full square.
Hexes have fewer issues with that kind of stuff. With 6 sides instead of 4, diagonal movement is easier to handle, as is diagonal attacking since every neighboring hex is the same distance from the hex you're standing on, while on a square grid the distance of diagonal squares is higher than the distance of straight squares. You don't have to make up any special rules such as "diagonal movement costs 1.5 AP" or "diagonal attacks are only possible for reach weapons". Everything just works better from the get-go, without any special rules required for diagonal things.
Hexes also make for more natural terrain design than squares, which is one of the reasons we went for hexes in Realms Beyond. Nature isn't square, and with a hex grid it's easier to create irregular terrain than with a square grid.
For tactics and area effect spells hexes are also better suited. We've made a nice graphic to illustrate it:
So overall, hexes seem to be the superior style of grid, offering higher tactical variety, easier handling of the diagonal axes, and a more natural and less "boxy" level layout.
However, that doesn't mean squares don't have advantages, too. Even though diagonal movement isn't as natural as with hexes, a square grid potentially offers 8 directions to walk into, where hexes offer only 6. Depending on how you deal with diagonal movement and attacks, square grids can potentially offer more tactical depth as far as maneuvering and outflanking is concerned.
Personally, I think the best system would be one of square tiles with a good and tactically interesting implementation of half-square tiles, essentially turning the grid into an octagonal one with small half-grids filling in the empty space between the diagonals.
Both squares and hexes are limited and boring, give me some fuckin OCTAGONS