OLED displays have a very wide color gamut. If you have source material that was tuned to it, there's really nothing like it. To get a similar gamut with LCD you need a quantum dot backlight. Samsung devices all have a screen mode where they constrain their gamut to sRGB for more accurate colors, though.
In terms of color accuracy, they've consistently been the best around, when set to limit their gamut to sRGB. This (http://www.displaymate.com/Galaxy_Note7_ShootOut_1.htm) site has test results for most recentish smartphones and tablets and declared the Note 7's display to be "the best performing Smartphone Display that we have ever tested." Every Samsung device with an OLED display released in the last few years has achieved this.
And of course the contrast ratio is much better than LCD. In terms of display quality, provided the phone can be set to sRGB mode, OLED is as good or better than LCD in every metric. And this doesn't even touch on all of the novel uses a display that uses no power when displaying black can be put to - for example, Samsung devices can typically display the time or notifications in white-on-black on the display persistently when the screen is otherwise 'off'. With OLED, only the pixels that are active are drawing power.
The only problem with OLED has been blue pixel decay. I have an OLED display in both my tablet and my phone and I'm a heavy user, and I haven't noticed any change in the 2 years I've owned them though.
Really, it's just a better technology. It's shameful that everyone has allowed Samsung to maintain a lock on it. I really wish I didn't have to buy Samsung devices to get this fantastic technology.
So just the entire Internet?
Seriously though, from the above linked article,
While LCDs remain more power efficient for images with mostly full screen white content (like all text screens on a white background, for example), OLEDs are more power efficient for typical mixed image content because they are emissive displays so their power varies with the Average Picture Level (average Brightness) of the image content over the entire screen. For OLEDs, Black pixels and sub-pixels don’t use any power so screens with Black backgrounds are very power efficient for OLEDs. For LCDs the display power is fixed and independent of image content. Currently, OLED displays are more power efficient than LCDs for Average Pictures Levels of 65 percent or less, and LCDs are more power efficient for Average Picture Levels above 65 percent. Since both technologies are continuing to improve their power efficiencies, the crossover will continue to change with time.
At peak brightness with a pure white screen, the iPhone 7's screen draws 1.08 watts, whereas the Galaxy Note 7's draws 1.8 watts. So if you keep your phone at peak brightness, I guess that's one metric where LCD beats OLED.
I should note that for some reason it seems like most smartphone manufacturers, basically everyone aside from Apple, way oversaturate their displays, this is especially true in the case of Samsung. Even the Nexus 6P display looked horrendous to me out of the box. Fortunately there is a setting in the developer options to enable sRGB mode which makes a huge difference.
Regarding power consumption, I think OLED mostly loses out on efficiency when generating white backgrounds, but I'm not sure how much of a battery hit this causes in normal use scenarios.
As to what's so great about OLED, for me, as someone who frequently uses their phone in relatively low light situations, I absolutely love the deep blacks and the fact that when I have the phone on low brightness using a dark theme it barely produces any ambient light. On that note, Google seriously needs to introduce a system wide dark theme. I don't have an iPhone but I'm really interested to see if/when Apple adopts OLED whether they redesign iOS to take advantage of it. I know many people dislike the bright white UI of iOS and Android.
Yup - completely agree. When I got my Tab S, I knew that the exynos 5420 was grossly inferior to the a8x in the Air 2, but I wanted my tablet for watching videos in bed at night.
I was right; the 5420 is execrable and makes web browsing very painful compared to the Air 2 (my kids have Air 2's). But the display is so much better than the Air 2's in every way you'd care to mention or measure. It's unbeatable for its intended purpose.
<sigh>
I'm sure with the large screen on a tablet this is even more of a benefit.