Why does the U.S. military maintain their own weather satellites like the DMSP Block 5D2?
What's the difference to "normal" weather satellites? Do they use other sensors or orbits?
If they have their own satellites, do they also have their own super computers and weather models? Why the effort?
Note: This answer is based on a source from 2001. It provides a lot of background as a historical overview, but it does not take into account recent changes in the program. Please review the end of Organic Marble's answer for a fuller perspective.
Initially, it began as a temporary program. In the late 1950s, the US military needed accurate and timely forecasts in order to plan times for cloud-free observation of Soviet targets with photoreconnaissance satellites. When a study of the subject completed in 1961, NASA was instructed to develop meteorological satellites for all civil and military forecasting needs, but the Pentagon remained unconvinced that the NASA program would meet the needs of the National Reconnaissance Program (NRP). They believed the NASA plans would not meet their requirements for secrecy and that the limited coverage they would provide would be insufficient. In July 1961, the first director of the DMSP was appointed.
By 1969, all three military services had begun to rely on the DMSP, and sought weather support on a daily basis. The satellites by this point met tactical support requirements and so was of utility for more than one purpose for the military. The program intended to be temporary had become permanent.
By mid-1980, though the NOAA satellites often provided better service, the combined data from NOAA and DMSP sources aided forecasting. In 1993, it was decided that the efforts be combined, and NOAA would hold overall responsibility for the joint system. In 1998, NOAA assumed responsibility for controlling the DMSP satellites.
(There is a lot more to the story. If you're interested, I would highly recommend reading the full document.)
The specific needs of the military may not be served by civilian weather satellites.
Specifically, the DMSP started as a classified program that supported the Corona spy satellite program. Its purpose was to predict cloud cover over foreign countries so that the expendable film in the Corona satellites would not be used up taking pictures of the tops of clouds. From there, as systems tend to, the requirements crept. Satellites fielded in the 1970s could be used for battlefield damage assessement.
In 1993 the DMSP had grown to where it was pretty redundant/parallel to civilian systems however, and it actually was merged with them.
In May 1998, the US Air Force Space Command transferred its day-to-day operations of the DMSP spacecraft to NOAA. With this action, NOAA assumed full responsibility for the operation of both the POES and DMSP satellite constellations.
Portions of the polar-orbiting meteorological satellite program were split out again in 2010 after the failure of the National Polar-orbiting Environmental Satellite System project.
Lots of good information at the source I used, the Earth Observation Portal DMSP page.
Not to be a conspiracy theorist, but a perfectly reasonable explanation is that they are actually used for surveillance purposes.
The military weather satellites have remained separate from the civilian satellites operated by the National Oceanographic and Atmospheric Administration (NOAA), at least in part because the ground terminals that pick up data from the satellites can be used to transmit "image" or "don't image" commands to the spy satellites. (page 82, Google books link)
Why? Because the US military wants to know the weather in warzones and possible future warzones (and not the USA), and that data may not be available from commercial sources. Commercial companies may be ordered by local government to restrict their data in case of war, etc.