Why do computers no longer use screen savers? Is it because new monitors have the ability to turn themselves off and in the past they were not able?
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Because the original reasons for using screen savers no longer apply.
The Wikipedia page on screensavers explains it, but older CRT screens were vulnerable to phosphor burn-in. In cases of prolonged static images on old screens, "ghost" images would remain even when new content appeared on screen.
LCDs and most other modern monitors are not susceptible to this phenomenon (though plasma screens and the current state of the art in OLEDs are), and thus screensavers are not necessary. Of course, they still survive because "it's always been that way", and once (if) OLEDs take over, screensavers will enjoy renewed relevance.
Nowadays, though, since we're more conscious of energy usage, it makes more sense to simply power down the monitor when it's not in use.
Partly because of power saving features, though those have existed for quite some time now. It's more due to the fact that LCD monitors aren't as prone to the "burn-in" that old CRT monitors had to worry about. If you left your old CRT on the same program for an extended amount of time(or used it every day), it would begin to burn after images onto the screen that could not be removed. Screen savers prevented this from happening as frequently by constantly altering the pixel arrangement.