I have both an Asus 60hz IPS monitor and a BenQ 144hz monitor. I had the Asus first, and use both currently.
Honestly, a lot of it will come down to personal preferences. After using my Asus monitor, I found out that I'm not picky about colors and I don't miss it while using my BenQ with a TN panel (note - I'm supposedly red-green colorblind). I do enjoy the higher refresh rate very much, even on mundane stuff like regular desktop use. Other people will take the improved color quality over refresh rate if they have to make a choice, I'm one who would choose higher refresh rate.
Speaking technically, one of the biggest advantages in gaming for a 144hz TN vs 60hz IPS isn't necessarily the higher refresh rate itself but the reduction in "motion" artifacts that results from the higher refresh rate. When I used to play battlefield 3/4 a lot I had only my Asus monitor most of that time. When I got the BenQ monitor and enabled the strobe backlight at 120hz, I was blown away by how much easier it was to see enemy players during fast paced matches. On my Asus monitor, the ghosting and pixel (motion) blur was noticeably worse. Read more about these artifacts here -
http://www.blurbusters.com/faq/lcd-motion-artifacts/
http://www.blurbusters.com/faq/60vs120vslb/
LCD artifacts aside, I can easily notice the difference in frame rate between 60hz and 120/144hz and prefer the latter. Games feel smoother and have snappier responses.
Of course, you'll need to be able to push >60fps to get the most out of a 120/144hz monitor (note - Even if you can only manage 60fps in a game on a 144hz monitor, you still get the benefit of reduced LCD artifacts). I've used both GTX 660ti and Radeon hd 7950 (equal to the R9 280) and had no trouble getting high frame rates at 1080P. I do tweak settings in every game I use though, typically I have 4x AA, 16x AF, Textures at high/ultra and everything else low/off.
Edit: I see we have some terminology to straighten out -
LCD: Liquid Crystal Display.
LED: A Liquid Crystal Display that uses an LED backlight instead of flourescent bulbs.
IPS: In-Plane Switching panel. It's one of a number of different kinds of panels used on LCDs.
TN: Twisted Nematic. Another type of panel for LCDs. 120hz/144hz monitors are usually TN. This is the most common panel type.
I may not have covered it well, so I'll clarify here - your in-game frame rate is very important in this discussion. A 60hz monitor can display a maximum of 60 frames per second, so even if you run a game at 100 fps you will only see 60 of those 100 frames. Likewise, on a 120hz monitor 120 fps becomes the limit. Running a game at less than 120fps on a 120hz monitor is still fine by my book, as you still benefit from reduced LCD artifacts like ghosting/blurring.