OLED panels currently have generally worse uniformity than LCD, including the current-gen Samsung phones. A gradient with black crush across the panel is extremely common. Many users don't notice it, but it's hard to find a unit that doesn't show this effect.
The current Samsung OLED panels also use PWM dimming, which means they control brightness by changing flicker rate instead of actually dimming. This can be irritating to the eyes and produces a stroke effect when the phone is moved, but again many people don't seem to notice. Apple is pretty serious about using flicker-free LCD panels but many other manufacturers use PWM with LCD as well. PWM is cheaper to implement, and in the case of OLED it may also reduce power to "flash" the screen at high brightness instead of regulating a constant output.
There are some minor issues with low brightness pixel response times which can be seen as purple trailing effects.
And most importantly the current batch of phone OLED panels are "pentile," which means their true resolution is not nearly as high as the advertised/rendered resolution. Some details in the Anandtech Note 4 review (still using the same pixel layout and resolution in the current panels):
http://www.anandtech.com/show/8613/the-samsung-galaxy-note-4-review/4
That pentile layout is generally considered to be a workaround for the uneven aging of the different colored subpixels on OLED, which brings up another point: Burn in.
My point isn't to say that these displays are in any way bad; They look amazing even with all these negatives. The point is simply that there ARE downsides to the technology in its current state. The real future of display tech is probably micro-LED, which uses the same idea (individual diode for each pixel) and uses miniaturized traditional LEDs which don't suffer from as much burn-in as OLED.