Let's say that we have same average quality LCD and CRT monitors. They have same size, same resolution and same frequency. Let's say almost the same everything (temp, room light, etc). Which would be the differences between them to the eyes?
TLDR: If you are on a very small budget, get a TN screen. If you are on a decent budget, get a nice budget IPS screen. If you're living off a shopping cart, go salvage a CRT and pray it doesn't blow up. If you want a massive screen to go with your massive mansion, go for a Plasma Display.
I'm referencing this Wikipedia article heavily and it's worth a read. In addition though, there's quite a few reliability related factors that would be in play here.
Let's start with the grand old man of display technology - CRTs - they are resolution independent and can handle any resolution and synchronization rate that is within the specs of the monitor - I can run a 1280x1024 (the highest resolution CRT I've owned) monitor at 800x600 or 1024x768 without it looking like garbage.
CRTs are simple devices that are complex to build - They light up pixels with an electron gun. As a result CRTs have incredible contrast (better than most LCDs), incredibly fast refresh rates and response times and very good black reproduction since they produce light, rather than blocking it out. If they fail though, good luck finding someone to repair them. They also refresh line by line, so unless you have a very high refresh rate, you would likely get eye strain (this also produces an awesome effect when you attempt to capture them on video. They don't make CRTs anymore and any attempts at modern CRTs like Field effect displays haven't panned out. They suffer from burn in if they are left for too long.
Plasma is a bit of a Rolls-Royce of display technologies - they have the advantages of CRTs, such as awesome black reproduction and response rates, MASSIVE screen sizes, but are really expensive. You'd likely end up buying a TV rather than a dedicated monitor. Do this if you want a MASSIVE high quality screen, and don't mind blowing the budget for it. They suffer from burn in, and double up as a space heater, since they produce images using ionized gas. Lovely things, beautiful images, but seriously impractical.
Practically? LCDs are your only real option these days. We can differentiate LCDs by panel types (which is what you should be looking at) and backlight.
Let's start with backlights - the classic LCDs used a high voltage cold cathode backlight. This needed an inverter, a fairly beefy power supply, and various parts of this could fail - I have a project monitor that has had a half dozen caps replaced so far, and is waiting for me to replace another one!
Modern LCDs use LED for backlight - this question covers the difference between WLED and RGBLED backlight - in short, RGBLED gives better colours since wled backlights have a yellow tint.
There's also panel types - the common panel is TN - inexpensive, good enough, but suffers from restricted viewing angles and other minor issues. The monitor you are using now is TN so if you're happy with it, the smaller colour depth and viewing angles are not an issue for you.
IPS or in plane switching is better - its what the cool kids with snappy monitors use - it has better colour quality, viewing angles and many of the disadvantages it had with respect to a traditional TN screen have been worked through - It was expensive until recently there's some good, relatively inexpensive panels these days.