A big reason why "sandbox" games like The Elder Scrolls or Grand Theft Auto are so popular is because they give you more freedom to play the game your own way. Even the most linear, script-heavy game benefits from adding bits of freedom to let the player customize their gaming experience. Video games are interactive, after all, and it can be really annoying when a game forces you to play a certain way when other options seem completely obvious or simply more natural.
The most important aspect of freedom (whether it be the freedom to do quests in any non-linear order, or merely the freedom to choose your skills and specializations in a linear story) is that the player's input has a significant impact on the gameplay. You really feel like you're in control of the game and that your actions matter. "The main quest wants me to do this, but I don't want to yet. I'll do it later," or "I could finish my objective the obvious way that the game wants me to, or I could devise a clever scheme to get it done more efficiently." It just feels rewarding to make your own decisions.
More after the jump.
But some games are so restrictive that you may as well just be watching a movie, or just pressing a single button to make the game progress forward. Everyone who plays that game is going to have the exact same experience as you, and it doesn't feel as personal. A lot of the fun from games with immense freedom is that each player has their own unique stories about what their adventure was like or what kind of situations they ran into. It builds a deeper connection with the game and prompts a lot of great discussion about the game.
The freedom to choose your gameplay also adds extra consequences for your actions, which further adds to the sense of reward when you overcome adversity or make a smart decision. Instead of merely having "success" or "failure" as options, you often wind up with a lot of in-betweens; you get the job done but at great cost to your own health or resources, or you get the job done flawlessly while killing an NPC whom you may want to have around later. If you develop a really strong build for your character, you can feel proud knowing that you did that yourself, especially since it might be totally possible to make a bad build that doesn't work as well.
But there is a problem with having too much freedom. I call it "Oblivion-syndrome," because there's just so much to do that you don't know where to start and everything begins to feel bland after a while. One major problem I rant into with Oblivion was that I'd pick up a quest, start to do it, and pick up six more quests in the meantime. The first quest brings me to a new area where I inevitably get curious and want to explore the surroundings, and inevitably end up picking up even more quests or getting involved in random encounters and other such things.
I ended up strung out too much trying to do too many different things, even getting caught up in things I didn't want to mess with. The amount of freedom is kind of like being stuck in the middle of an ocean, treading water with no land in sight. You can go in any direction you want, but it's all going to be the same.
Portal 2 is a case of both good and bad levels of freedom. One the one hand you have extremely long sections where all you can do is follow a tight, narrow hallway and sit through scripted events that happen the same regardless of what you do, doing simple puzzles where all you do is look for two isolated patches of portal walls. The scenery is nice to look at it, but it would've helped to have been able to explore off the main path a little bit, or to have alternate ways to get from point A to point B.
But on the other hand, you have the test chambers themselves, which present you with a puzzle that you're free to solve as you see fit. Even though there's ultimately only one possible solution (and I rage about my ingenious methodology being pre-emptively prevented), it can be pretty fun to experiment with things on your own, trying different things until you figure out what you have to do. It's a case where an extremely linear game can still give you freedom to make it feel more interactive and rewarding.