If you have natural terrain (fields, caves, etc), you don't really run into the downsides of hex grids. They often look better there, since right angles are not common in nature. they look worse in most man-made areas, though.
The hex grid distance metric is much closer to euclidian distance than any of the reasonable metrics for square grids. This make them superior in tactics games. With a square grid, you have to do strange math things if you want the space to feel like euclidean space (you have to make diagonals work, and it's hard to do so with integer-logic). On the other hand with hex grids the player's intuition about distance is pretty close to the in-game distance.
In a square grid there are only 4 directions you can move that go through an edge. The other 4 go through corners. This can cause issues in games when diagonal movement or range counting is allowed as technically 15 squares diagonally is longer than 15 squares horizontally or vertically (-> see Pythagoras).
Hexagons on the other hand have 6 degrees of freedom and all movement is through an edge, never a corner. This makes range counting easier and more consistent as well.
However it's hard to draw right angles in a hex grid (i.e. if you take the 3 axis approach, there's others, then no axis is at 90° to another). That's why square grids are more common for dungeon style maps with straight corridors and right angle turns while hex grids show up more for over world maps where sharp corners are less likely.
But there's a solution for everything:
Just use
Kisrhombille tiling and you can use hexes and draw straight lines!
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal works best.
It's Darkzone, one of the Wasteland 2 official forum "celebrities".
