Building from scratch vs. buying new
The pros of buying pre-built are the cons of building your own.
Each approach is born of very different requirements.  That is, the person who is actually likely to build their own is very unlikely to buy off the shelf and vice versa.  (I assume that 'build from scratch' also covers getting the LBS to do it for you to your specification).
I would caution everybody thinking of building their own that you will get much better 'value for money' if you buy pre-built.  A $1000 pre-built will get you much higher quality parts than a $1000 build your own.  This disparity seems to reduce the sillier the price becomes at the high end.
You will need access to some specialized and expensive tools (thanks Gary!) and the skills to go with them.
There is, as always, an intermediate stage where you source everything yourself but have the LBS do the trickier jobs like headset pressing, bottom bracket cups and steerer tube cutting/star nut pressing.  Just remember that your LBS is a business and needs to make money - so buy the parts from them if you want them to fit them!
Buying New
Pros

Integrated package
Lowest cost
Guaranteed that all the parts will fit and work together
Just go pick it up from the bike shop and you can ride
Don't need to know as much about bikes
Possibility to test ride (usefulness varies, often takes a while to get to know a new bike)

Cons

No choice of components
The color frame you want is either too cheap or too expensive!
Often component compromises are made to reach a price point
You don't get to choose what factors to optimize for

Build Your Own
Pros

Get exactly what components you want
You can choose what to optimize for
You can get your frame color
You choose what compromises you make within your budget
Every part is chosen for a reason and you get to spend hours researching it
The incredibly deep satisfaction you feel building a bike from scratch
Knowing that nobody else is riding a bike like yours
You may need some very specialized (read expensive) tools and you end up with the ability to use them.

Cons

You spent hours researching every part
Building wheels from scratch is not for the faint hearted, but it is rewarding
It's the most expensive way I can think of to get a bicycle
You may be more prone to 'overprotective parent syndrome'
You may need some very specialized (read expensive) tools and the knowledge to use them
You must be comfortable filing and sanding and abusing your frame (think headsets and bottom brackets)
You will make mistakes with part compatibility
It takes a lot longer to get riding
Only feel how the bike rides after spending a lot of time end money, although you can often find people with similar bikes to test ride (see above)