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that horizontal tearing is from NOT using vsync. Unless you have a very high end system nad aren't playing the latest games, i would say leave it off. If you cant keep frame rates as high as your refresh rate, then they drop by half. For example, if you are running at 60hz, and your card can only pull off 57fps, your frames will drop to 30fps... and god forbid, if you get in a shootout while frames hit 29fps, your fps will go to 15fps... not good
Hopefully you can find all the information about tearing and V-Sync here in this guide. Answers a lot of questions, and has a lot of facts in it. Read it, and find out what you would prefer.
EDIT: Noted exactly what the problem is. It is more than likely that the tearing on the screen comes from your system running the game at a faster fps than the refresh rate, ie V-Sync is not turned on. Again, read the guide above for all the technical stuff about how that works. Generally speaking, you should keep it off if you know that your comp won't be able to produce a framerate to match your refresh rate in that game. If it can easily produce that framerate, and doesn't go below it, AND the tearing is bothering you, then turn V-Sync on.
That is weird. I have problems in games with V-sync off.. when my framerate went above the refresh rate, it actually became choppy.. With V-sync on I just max out at frames=refresh rate, but it doesnt really affect framerate otherwise.