The advantage of Limited Use power-ups is that it gives the level designer the ability to have almost unlimited control over how the player approaches a stage. For example, the game gives the player the ability to fly using a certain power-up. Flight is always an extremely powerful ability. If a player is allowed to fly at any point in the game, it might become possible for the player to bypass most obstacles or even reach places where the designers didn't want him to reach. However, if the player can only fly by collecting a certain item that appears in a specific place and limiting how long the player can use the power, then the designers can prevent the power from becoming a problem. Movement powers such as high jumps or flight tend to be the most common examples of powers that are limited.
However, while the design philosophy of Limited Use has its uses, I have never liked it that much. First off, if most powers in a game are designed to be Limited-Use, then it often becomes obvious as soon as the pertinent power-up appears that you need that power in order to progress through the stage. For example, if you find a flight pad and Red Feathers in Banjo-Kazooie, you obviously need to fly (usually along a route that has Red Feathers) in order to get to your objective. While this can be a useful tool for helping players along in easy games, it can be very predictable. I find it boring stage design myself. In some cases, such as the use of limiting abilities such that they can only be used on specific pads, it can become intrusive to the game and obnoxious.
The greater problem with Limited-Use is that it prevents the player from having fun just paying around with powers. Sometimes it is just fun to be able to play around with an ability, or to use it in unexpected areas to see if you can find something unexpected of hidden. Often, when there are powers that are limited use, the game designers won't make them usable except in places where they are essential to complete a task. However, this means that designers will often to forget to give the player access to some abilities most of the time. For example, Fire Flowers are apparently a very rare thing in Super Mario Galaxy. However, they would not cause any problems in the form of letting the player get into places he should not be, nor would it upset the difficulty of the game too much. Their rareness is inexplicable, and is a bit of a lost opportunity.
The reason I prefer Free Use design is that it opens up the possibility of the player having more than one way through an area. For example, say that a certain power-up can be found in Stage A, but not in stage B. If the player can manage to not lose the power-up after clearing Stage A, he might be able to find a short-cut or hidden reward in Stage B using it. In this sense, Free Use design opens up more design space and rewards the player for playing around with an items powers in various areas. There are actually quite a few examples of this in various Super Mario Bros. games.
While Limited Use does make design work easier, I think Free Use creates a more rewarding game experience. While it is easier for the designer if you don't have to worry about a player jumping or flying over a wall you don't want him to cross, there are always ways to solve these problems without eliminating the freedom of the player.