With games like the Dresden Files and other FATE systems starting to take over the concept of a "party" is starting to become a lot more optional than I would have guessed a few years ago. Sure, I knew you could - and at times did - run games where the players weren't a formed party, but the default seemed like it would stay as a party of adventurers getting into adventures. Today I want to talk about the strengths, and weaknesses, both approaches can have. Who knows, maybe it'll help you with your own games.
Party of Adventurers - Pros
The pros to a party of adventurers are well known and some may seem blatantly obvious. For one, you have the players all grouped up and working together towards a common goal. This can minimize PvP gameplay and keep everything focused. The party, being a party, also helps the players relax and have fun since they're "all in it together" in a matter of speaking. The party shines in other ways as well.
Plot wise, it is easier to do big plots with the party. You can do truly globe-shaking plot lines almost with impunity and have the adventurers work their way through it. Challenge assessment is also easier since the group functions as a group and can be treated as one entity. In some games, like D&D, this is almost formulaic. You have 4-6 level 10 adventurers so flip your Monster Manual open to the Challenge Rating 12 monsters and voila, instant challenge custom leveled to what your party should be able to handle.
Less martial challenges are also easier since odds are someone in the group will have what is needed for a given task, or be up for role playing their way through the awkward conversation with the King's inquisitioner.
Generally, you get the benefit of working with a team. Even if it is a team of shambly shiftless layabout barbarians.
Slit Group - Pros
On the other hand you have the split groups. These people, while willing to work together at times, for the most part are involved in their own affairs and their own threadlines. They move, often through a set part of the setting - like a city or castle - and handle their own business while brushing up against or otherwise interfering int he business of others. This type of setup works wonders for any political style game and lends itself well to some PvP provided the group is mature enough to handle it.
Where the split group shines though is personal stories. Everything gets pulled down a level scale wise, but gets pushed up several levels on what things mean to the character involved. You may not be saving the world, but you will be exploring what your character really wants and what they are willing to go through to get it. Personal possessions, even loved ones, can all be fair game in these kinds of stories and that ratchets up the intensity, drama, and fun (with the right group, anyhow) in a way that saving the world of disarming a bomb with only seconds to spare simply can't compare to.
The strengths of one are mostly the weaknesses of the other. For example, in a Party challenges are easier on the GM because someone in the group should have the skillset to handle it. In a split group, challenges often have to be custom built for the character in question, or to lead them to needing outside help, or you find yourself - and the player - at a dead end. Another example would be that party games often don't do well at personal plot lines (the other characters interfere too much) while split groups often don't handle big events (things go slowly, and they backstab themselves at times.) Both can be done, but they shine in their own areas.
That last sentence there is key. Don't let the fact that you have a party game stop you from doing personal plot lines. Also, don't let a split group stop you from threatening the world. Just understand that those games may have hiccups and road bumps with those plot lines that you aren't expecting. After all, saving the world is all fine and easy until someone knifes the fighter in the back before he can fight the enemy general.