3 piece necks are stiffer and tend to be brighter.The 3 pieces generally run lengthwise with the neck.
If you have a 1 piece neck made out of good, stable wood, then it should be strong enough. However, to made the neck even stronger and resistant to warping, a guitar manufacturer, such as Hamer, may cut that same piece into 3, take the middle piece, reverse it, so that it opposes the grain of the other 2 outside pieces, making the neck stronger.
However, some guitar manufacturers (like Gibson in the 70's), either run out of high quality wood to make a stable 1 piece neck, use cheaper wood that would normally warp, but if it is laminated into 3, the opposing grain patters strengthen the neck to make the neck usuable. This is done on doem low end Ibanez guitars. Ibanez used to use solid quartersawn maple on their super thin Wizard necks, but with time switched to a 3 or five piece construction, to make a neck that was just as strong, but cheaper to make as they used cheaper wood.