Do better board representations than the standard one exist? Consider the classical representation of the state of a chess game. I'll draw the board:

R BK BNR
PPPP  PP
  N Q
    PP
    p
  np p
ppp   pp
r bkqbnr    


Here is its move list:

d5 d5
c4 c6
Nf3 e6
Qd3 Nf6
...        


Here is its FEN notation:

rnbqkb1r/pp3ppp/2p1pn2/3p4/2PP4/3Q1N2/PP2PPPP/RNB1KB1R w KQkq -


These three ways to describe the game state are isomorphic and contain exactly the same information represented differently. A computer would have no problem converting from one representation to the other (kind of, the castling flag isn't available in the first representation and the second one encodes all the game states). Computers uses even stranger representations like bit boards which I won't go into here.

In fact, there is an almost endless amount of ways to represent the state of a chess game. For example, here is one I just came up with:

R.BK.BNRPPPP. PP..N.Q.......PP......p.....np.p..ppp...ppr.bkqbnr    


But if I asked you to "find the best move for white" the first representation would be much easier for you than the others. 

Why? Is it just because the classical representation is what most players are used to or does it have some inherent advantage? Could you train your skill at playing chess using only FEN notation and over time come to prefer that? Has any research been done into this area? Maybe there is a better way to represent a chess game than the classical one which gives you a huge advantage when calculating lines deeply but just no one has found it?
Chess is a game in two dimensions. The game is made to be played in two dimensions. Pieces move horizontally, vertically, and diagonally very naturally on a chess board. It's purely visual and the sort of thing our brain does very well. Other depictions of the position sacrifice the two dimensional nature of the game. When this information is lost, we have to calculate to reconstruct the spacial relationships in the game. Consider the difficulty of spotting a pin or discovered check when using FEN.

In the other notations other than the standard diagram, you'll spend a lot of time doing mechanical sorts of math.

The different representations (diagram, moves, and FEN) exist because they fill different needs.