I want to know what difference does it make, in terms of observations made and the benefits for the astronomical society if we place a space telescope after the asteroid belt or near the Kuiper belt. I do understand it's either immensely expensive and technologically may not be feasible yet.
Very little upside, lots of downside.
It's colder out there, which means that keeping your IR sensors cool is a bit easier.
The orbit is wider, so you get more parallax, but the orbit is slower, so you have to wait longer to get the results.
The sun is smaller and dimmer, so the area of the sky that you have to avoid (as it is too close to the sun) is smaller.
The downsides are much more significant:
You get less telescope per dollar. For a given budget your telescope must be lighter, smaller with lower resolution.
If something goes wrong, you can't fix it. You can send a rocket up to fit corrective lenses to Hubble; you can't do that if it is out beyond the asteroid belt.
If you put it in an ecliptic orbit in the plane of the planets, then your probe will go behind the sun every year. While it is behind the sun, you can't communicate with it.
If you put it in some other orbit it takes much more fuel (and so you get an even smaller telescope for your money)
Data transmission rates are much lower. Its acceptable for New Horizons to spend a year to download all its data from one day observing. But if you only had one observing day on your telescope per year that would be disappointing.
All in all this is a "why don't we build houses out of gold". There might be a few minor advantages (gold doesn't rust) but the disadvantages (gold is costly and weak) far outweigh the advantages.
I don't see the down sides to be anywhere nearly as strong as @JamesK's answer suggests!
Some space telescopes use several kW of power but one of them is 500 W and these are all circa 1 AU where power is plentiful so there was no need to lower it. In deep space if you really need to just collect CCD images in cold space and process them with processors available in 2020 this could probably be done with much less power than these 20th century 1 AU telescopes used. Look at the processing power available in handheld devices these days!
Telescopes like GAIA spin constantly while collecting data and uses a phased array antenna to communicate while rotating, so spin-stabilization will be less power-hungry than some alternatives, there is a lot of room to optimize the design for low average power.
You don't need a bigger antenna or more powerful transmitter because you'd use an optical communications link and a 1 Watt laser diode. You'd deploy a separate, non-spinning smallsat to do that and connect by WiFi.
Here is a link budget calculation for an optical versus radio-based deep space link, and another mention of a next-generation deep-space telescope.
The cold will be a big power savings; you won't need refrigerators or an attitude control system to keep a sunshade pointed towards Earth.
Probably a 100 or 200 W RTG is all that's necessary to run a decent space telescope far from the Sun using technology from 2020.
So all of the merits pointed out in the other answer are really available and worth exploring in greater detail in yet another answer here rather than dismissing the question so out of hand!
Copied from Wikipedia articles:
Telescope Launched Power (W) --------- ------- Spitzer 2003 427* WISE 2009 550 Herschel 2009 1000 Kepler 2009 1100 GAIA 2013 2010 Hubble 1990 2800 *http://www.spitzer.caltech.edu/mission/188-The-Solar-Panel-Assembly
While technically doable, there's substnatial risk involved and little perceived benefit.
- It would take YEARS to make it that far.
- Being that far from the Sun, means solar power is not an option for utilization as a power source. That leaves RTG's, but they produce far too little power to run a large telescope (The Hubble uses 2.1 kilowatts of power) A telescope Hubble sized would need 10 RTG's to run
- Being that far from Earth makes repair and servicing impossible.
- The Telescope would require a extensive and expensive monitoring network to retrieve data and imagery.