(Answering the question introduced by Brian's title edit)
DAS and DIP are both tools, and as such they each apply to slightly different situations. They also have quite a lot of overlap: DIP characters can often benefit from having a DAS skeleton, while DAS can benefit from being tweaked through the first few sessions of play.
Before getting further, it's important to note that DAS and DIP are both extremely unimportant compared to character development. While a skilled DM will weave your character's chosen personality and background into their narrative, what you did in the actual campaign proper will be remembered long after your written backstory is forgotten; how you reacted when you were sitting at the table will make a stronger impression than how your character sheet says you should have reacted.
Character creation is a starting point -- A means to begin the character creation process.
The primary strength of DAS is that it facilitates communication between the player and the GM. It tells the GM exactly what hooks exist in your character's past, and spotlights what you're interested in following up on. It gives the DM something they can refer back to throughout the campaign, and gives them a better chance of surprising you.
The other big advantage of DAS is that it allows you to enforce consistency on your group. People play RPGs for a variety of reasons. DAS allows you to cull players with a different outlook on the hobby quickly, based on what their character descriptions sound like (or if they exist at all).
The D&D 4e DMG actually has a pretty interesting analysis of different types of players. I highly recommend to DMs frustrated with people who are playing the game wrong.
Finally, DAS helps you analyze your own patterns and push your abilities. It's a lot easier to stretch your acting ability if you have a written goal. And it's a lot easier to notice when your characters are blending together when you have written back stories to analyze.
DIP helps create a more organic party. The typical DAS workflow is as follows:
The DM describes the campaign in very loose (and often ultimately inaccurate) terms.
The players work out in loose terms who's doing what ("I'll play the healer").
Everyone goes home, and writes their own independent background and personality.
The DM gets a copy of everyone's personality and background (if they're lucky), and writes the introductory adventure.
There's no real emphasis on group dynamics. The world, and all members of the party, are each created in a vacuum by different people. How the personalities interact with each other is rarely, if ever, addressed.
By contrast, DIP is a collaborative process. Emphasize what gets the best reaction from the group. Downplay what causes tension. Even if players aren't specifically consulting each other, they do still base the end result on the actual realities of the gaming table.
The other main advantage to DIP is that it makes it harder to over reach. With DAS, it is very easy to create a character that you're not comfortable playing. The result of this, in most cases, is a character with little or no personality (it doesn't count if it's only on the sheet). With DIP, you'll quickly spot where you're hitting your limits, and adjust to compensate.
Finally, DIP is very non-intrusive. Many players will balk at writing a lengthy character description, but can be coaxed into developing a character through play. It allows the player who WANTS to write the elaborate backstory to do so, while the player who's mostly along for the ride (or to kill monsters and take their stuff) can go about their game in peace.
How I do things
Like most players, I mix and match. I'm a very flavor-oriented player when it comes to developing new characters, so I usually start with something out of the books that seems cool (class or archetype).
I come up with a sketch of the character in my head as I'm filling out the sheet. When I'm done, I give the DM the 10,000 foot overview of my thoughts on the character (mechanically and personality-wise).
Most of the character's actual personality then comes from the first few sessions of gaming. I start with sketch I worked out during character creation, but modify it to whatever seems interesting.
If the flavor and personality of the character particularly interest me, then the DM gets a written background a few weeks after starting.
How much DAS is involved for me also varies depending on the nature of the campaign. Starting characters tend to be pretty incompetent in most systems, so elaborate background stories don't make a lot of sense. On the other hand, characters starting well beyond the default starting point in the books need some background to explain exactly how they got to be so good at what they do.