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There are two common ways in which a game idea is born. There is the one where someone says: "I have an idea about a game where X character wakes up in a place and does this and that and is driven into Y adventure" (story driven game) or the one where someone says: "I have an idea about a game where X character must jump and throw stuff in this manner in order to achieve stuff" (gameplay driven game).
I must say I defend the importance of story telling in game design because it is a strong factor which motivates players to finish the game. Also, it is important for a game to be fun and that's why gameplay can't be ignored either.
So depending on the starting idea for a game there are differences in the outcome according to the features each paradigm adds to the game. If the game is story driven then it will probably have a strong initiative factor for the player to start playing the game (if the story has a good start) and also a strong motivational factor to keep the player playing until the end (if the plot is interesting). At the end of the story, the game may also have a strong publicity because the player will be recommending the game to his friends for its really great ending.
On the disadvantages of making a story driven game you will find that there are some problems if the designer can't match the story to an original game mechanic. Story driven games are characterized for its dynamic change in its game mechanics so they can add to the story plot.
Examples of this are games where you find items, unlock special skills and abilities, find special enemies, meet special allies (and maybe they can join you). If you limit the game to a game mechanic with few or no evolution at all then it would be probably better to just make a movie or write a book about it.
Another disadvantage of story driven games is that there are some players that just like to play for the sake of doing something fun (casual gamers). Some of these players are very religious about making the game experience casual and prefer not to get involved in the long term intentions of the game itself (the story).
A game designer must therefore be careful about choosing the pace of the story to match the target player, it is better not to leave the player waiting for special events in story to happen in order to motivate him to play more.
On the other hand there is the gameplay driven design which will be probably a light engaging game. This means that the game will motivate the player to play the game for the fun itself. If the player ever gets tired (which will probably happen) then the game will be over.
The design of a gameplay driven game is somehow faster than a story driven game: the designer must only think about the main game mechanic (the initial idea) and develop it in order to adjust it to some difficulty management along the game experience. Some designers prefer not leaving everything to just some difficulty tweaks and create a simple story ocurring along with the evolution of the main game mechanics.
Assuming the gameplay idea is very good (and is more probable for this to happen than in a story driven game) then there is a great advantage that can be taken into account for games developed with a game mechanic in mind from the start: it is easier to create a story that matches a game mechanic than creating a game mechanic which matches a story. It is also easier to adding the fun factor to a game when the gameplay is being thought from the start.