Although the .22LR and the .22WMR use the exact same diameter bullets the outside diameter of the cases are different!
The outside diameter of the .22lr is the same diameter as the .22 bullet. It is called an outside lubricated heel-based bullet because the part of the bullet that goes into the case matches the inside diameter of the case and is smaller in diameter than the rest of the bullet. That part is called the "heel"
Also, since this "heel" never touches the rifling of the barrel the lubricant is on the outside of the bullet where as you probably have noticed it can get dirty. Also, the case does not hold the bullet as securely as with most center fire ammo.
When they developed the .22WMR they wanted the bullet to use lubricant that was inside the case so it wouldn't get dirty and they wanted the case to hold the bullet more securely so they increased the overall diameter of the case so that the inside diameter of the case was .22" just like the bullet.
That means that the chambers of firearms that shoot .22WMR are greater in diameter than the chambers that shoot .22LR.
That is why some .22 rimfire revolvers come with two cylinders. One for .22LR and the other for .22WMR.