First, let's have a little lesson.
A turbocharger IS a supercharger. A supercharger compresses the intake air to provide more oxygen so you can burn more fuel. As simple as that.
The difference in the two is the way the compressor is driven. A turbocharger is a special kind of supercharger that uses the exhaust gas to drive a radial turbine compressor.
What people are talking about above are their experiences in the execution of the idea. There are many poorly executed designs out there. You can make a supercharger that has the high end boost like a turbo and you can make a turbo with NO lag like a supercharger - and there are plenty of both out there.
Since a turbo is ALWAYS remote mounted, it is easier to intercool it to boost that oxygen density even further. As mentioned above, though, that comes at the expense of complex exhaust routing.
Typically, a supercharger is easier to install, but does have some parasitic drag and, for the intake mounted versions especially, they are harder to intercool.
Because of the high compression ratios diesel engines operate at, they have a lot of energy in their exhaust streams which are "ripe pickings" for a turbo. From a factory standpoint, turbos have the advantage of quieting the exhaust and reducing the exhaust temp which helps them meet EPA requirements.
http://auto.howstuffworks.com/supercharger.htm