This was the first thing I thought about when I read the OP. There's room for a ton of differences in the telescope design, like what wavelengths the sensors are looking for. And how the scopes can be maneuvered and aimed.

There's no reason to think that these would be NASA's dream telescopes, even if by some definitions they are more advanced instruments.

That's what has me curious. The Hubble isn't designed to point at earth and would be of little use if we tried. Why does the US Military have telescopes that are on par with Hubble? Are they looking out into space as well? Why would they do that?

As far as the telescope optics is concerned looking up or looking down is irrelevant- 500km or 500 million light years is the same focus setting!

Looking up has more stringent pointing and tracking requirements just because we want to sit on the same object for hours at a time, but we also have lots of bright stars in the field to track. Although our downward looking colleagues also use small star tracking telescopes on their toys.

The big difference was generally in the cameras. Astronomers use a 2D CCD (like your digital camera) to take a long (hours) exposure which is read out at the end. Spy satelites (used to) use a 1D sensor like a scanner or fax machine which was constantly read out as the earth passed underneath - producing a long continuous strip image across a target.