Chess theory suggests that pawns whose file is nearer to the central files are more powerful. This is somewhat obvious for the rook's pawn, which can attack only one square as opposed to the other pawns, which can each attack two squares. However, in practice, even the rook's pawn turns out to be somewhat useful as a pawn at a3 or h3 can deter the opponent bishop coming to b4 or g4. In practice, are the pawns on central files really more powerful than the pawns on the side files? What is the reason? Does the situation change during the end game as opposed to the openings?
The more pieces on the board, the more powerful the center pawns. This is typical for most opening and middlegame positions. The most convenient place for pieces to travel is usually through the center of the board, and the center pawns help to control that area.
The less pieces on the board, the more powerful the wing pawns. Less pieces on the board means less need to control pieces travelling through the center. A passed wing pawn is harder to stop than a passed center pawn. Your opponent needs to go further to the edge to defend against it. This can really stretch out and discoordinate his pieces if he has to defend against another weakness on the other side of the board. The outside passed pawn is like a decoy.
Generally speaking, you can think of Center pawns becoming weaker and weaker as pieces are exchanged, and wing pawns stronger the closer you reach an endgame.
I think that powerful should not be mixed with valuable. The value of a pawn is relative to the concrete position on the board. The a-pawn or h-pawn might play a minor role in the middlegame. While in the endgame, they are valuable assets, especially when they become passed pawns and start marching towards their promotion. More importantly than exactly what pawns you have is the pawn structure, i.e. how are the pawns placed relative to one another. The number of pawn islands and the number of weaknesses will determine their total worth. I don't see a pawn getting weaker or stronger just because you are in the opening, middlegame or endgame. A pawn is generally said to become more valuable, the closer it gets to its promotion square. Pawns that collaborate together can strengthen each other and be powerful in attacks, see for instance positions with opposite castling (e.g. white castled O-O and black O-O-O). Now, looking at practical games, grabbing the center is a great way of fighting for the initiative. Attacking players prefer to do this as soon as possible. I think that this gambit against the Sicilian is an excellent example (white sacrifices the b-pawn to divert black's c-pawn from controlling the d4-square)
[FEN ""] 1.e4 c5 2.b4 cxb4 3.a3 bxa3 4.Bxa3
where white has 4 center pawns (c,d,e,f) against black's three (d,e,f). Also, white is ahead in development. All in all, white has the initiative.
Central squares are the key.
Occupying central squares is important in the opening and middle-game levels in chess. Pawns which are nearer to central files can attack the 4 central squares and protect own pieces there.
Side pawns are far from this battle, they should be unmoved until to choose the castling side, and after castling you shouldn't change the form of king's protector pawns as it's possible.