The main reason is that Cassegrain telescopes are much shorter than Newtonians for the same focal length. This means that the overall cost of the observatory, which includes a mount, dome, and building, will be much lower.
A secondary reason is that professional telescopes use heavy detectors at the focus of the telescope. Newtonian telescopes put the focus on the side of the telescope, so heavy instruments and detectors would need to be placed on the side of the telescope; in general, this can make things difficult, because the weight on the focus exerts a variable torque on the telescope's tube.
It's much easier to support these heavy instruments if they are along the optical axis rather than on the side of the tube.
Extremely heavy instruments like very large spectrographs sometimes require different, more specialized telescope designs like the Nasmyth focus.
As for your question on different types of Cassegrain - this is a nomenclatural issue and I'm sure there are many answers, but some I can think of: Nasmyth telescopes are sometimes called Nasmyth-Cassegrains. Occasionally, people lump Gregorians in with Cassegrains. Schmidt Telescopes use a spherical rather than parabolic mirror, which takes some corrective optics but provides a much larger field of view - as much as six to eight degrees, so they are used for surveys frequently. Like I said - I'm sure there are tons more.
There are lots of mathematical answers to this question, but I'd like to make a few qualitative observations instead, based on 54 years using telescopes of all kinds and sizes, from 40mm refractors to 74-inch reflectors.
- Unless you have some specialized purpose, don't consider anything
smaller than 6 inches aperture. Small telescopes look cute, but don't
show you much, especially if you're a beginner. Experienced observers
can tease amazing observations out of tiny scopes, but most of us
will be happier to give these a pass. Aperture wins.
- A 10-inch Newtonian on a Dobsonian mount is something of a "sweet
spot." It's about the smallest aperture to show significant detail in
deep sky objects, yet is compact and light enough to be easily
transported to dark sky sites.
- Above 10-inches, the more aperture the better, provided you can
comfortably transport, set up, and operate it. This is crucial! The
nicest telescope in the world is useless if it never gets used. I
find even a 12-inch Dob becomes bulky, cumbersome, and heavy.
Aperture wins, but only if you use it.
Best Answer
I think the first sentence from the Wikipedia article on Ritchey–Chrétien telescopes is one of the major compelling reasons:
Elimination of optical abberations is very important in the RC design. In addition to eliminating spherical abberations, which all cassegrain telescopes do, a RCT eliminates coma. This helps to maintain image quality across a larger field of view allowing for larger detectors.
In addition, it has a flat focal plane. This too is important for large field imagers since a lack of a flat focal plane means that if you focus the central area the edges will be out of focus and vice versa. This was an issue for old photographic plate systems as you would actually have to bend the glass of the plates to get the entire image in focus. Clyde Tombaugh used to tell a story about observing while looking for Pluto where he was out in the dome putting in a new photographic plate, and it shattered just after he finished applying the pressure to curve the plate into the focal plane of the telescope he was using. It was a cold night and he was afraid something more important had shattered.
The RCT design still suffers from astigmatism and field distortion as you move off-axis but it does manage to correct three of the five major abberations. So this type of telescopes is preferred for professional systems because it has a better optical image compared to other designs and allows for larger field of view.