Another subject with many sides to the discussion.
Everyone appreciates a good story, and you can tell the difference between a good and a bad one quite easily in games just as you can with films. Obviously not every game needs a strong storyline, like kids games for example and physically interactive games like guitar hero and singstar.
Otherwise, the story is really the driving force of games.
The game starts. Where are you? why? Who are you? What's your characters background? What happened to them? Where are you going? why? And so on. Everything that happens to the character is relevant to the story. This is true in a linear plot (e.g.Gears of war) but not always in more open games like RPGs as you get mini stories/quests where you can deviate from the main story line (e.g. oblivionIV).
Games can still flourish through attributes other than story, like how you control the character and their movements (e.g. Assassins creed), stylisation like Team Fortress and online multiplayer capabilities like Left4dead.
Companies tend to skimp on the writing because of the budget they have, without realising how important a gripping story could be to their game.
Personally, the story is an important aspect to me. I love to read a good fiction book and be totally immersed in the world. The same goes for games. You know when you get completely attatched to your character and you're focused and there's nothing around to distract. I want to react to the storyline, to have moments of shock, disbelief, revenge, sympathy etc. You can't get these emotions from a completely predictable storyline.
As for whether a strong storyline makes for a better game. Well, it makes for a better story and might keep you playing for a bit longer but it won't make boring environments, repetitive gameplay or lackluster characters any better, no. As I've said in my previous blogs, everything needs to come together and work in harmony. The storyline is just a larger piece of the jigsaw.