The Hz rating on your monitor determines how many times your screen refreshes. Since your eyes process images different from how your screen displays them, your eyes can occasionally catch the gaps in the display. The faster the refresh rate, the less likely you are to see those gaps. So having a higher Hz (Hertz) rating gives you an image with less flicker.
Additionally, the visual artifact known as "tearing" can be curbed by turning on "Wait for Vertical Sync" commonly known as vsync. This limits your FPS (rendered frames of animation per second) to your Hz rating. Naturally, if you take a hit of 30 fps during certain effects, it might not be so noticable at 85Hz rather than 60.
60hz is fine. It just tells you how fast the monitor flashes in 1 second. 60 is okay, you will hurt your eyes if you look at it too long (your eyes will hurt, it won't make you go blind) there are better. but 60 is fine.
My monitor is set at 60Hz and it works just fine. My eyes only hurt if I look at it as soon as i wake up or if I'm really tired. And usually monitors can go higher than 60. Mine goes up to 75hz i think.
For a CRT, yes it does matter. Read Cucumba's post about why.
For an LCD, it doesn't matter as much. LCDs don't flicker, they just change color/brightness. But, having a higher refresh rate on an LCD can smooth out things if the LCD can keep up with the demand of change. A slow LCD may only be able to change the color of its pixels 55 times a second, so anything over 60hz wouldn't be noticed. If your LCD can change the color of its pixels 75 times a second, then 60hz-75hz would be noticable.
It improves visual quality and lessens the chance of your computer getting 'tearing' during 3d apps. it also enhances your resolution changes (1280x1024@60hz is very different than 1280x1024@100hz). The refresh rate cooperates directly with your video card, so as I see it, it would be best to keep it as high as safely possible at all times.
60hz will make apps with v-sync lock at 60fps (like HL1 defaults to, for example). 60fps is great and all but when you could have as many as 100/120 its very foolish to leave it that low, if you ask me--especially since it burns your eyes, hurts your eyesight, causes migraines, and affects performance.