On the way to capture a pawn, the king must also keep the enemy king off its return route; a single straight-line move can throw the win away.
This is the position from Schlage–Ahues (Berlin 1921) after 1.Ke6 Kc3, the moment at which the player with White went wrong. White's pawn on a6 is blocked by Black's a7-pawn; the white king is on e6 and the black king on c3. White's plan is obvious: walk to b7, capture on a7, and promote the a-pawn. The difficulty is that after Kxa7 the king stands in front of its own rook pawn, so it must then get back to b7 or b8 before the black king reaches c7 or c8.