They work on completely different technologies and thus have the "opposite problems."
LCD works via a screendoor effect you have a white backlight and then you the lcd subpixel crystals to make that white light into blue, green, red, etc. Thus all LCD can never get perfect black. Including IPS LCD.
AMOLED has better blacks
AMOLED work by having each individual subpixel create its own light. Thus you can have a complete black by turning all three or four subpixels off with AMOLED. The problem with AMOLED is that the subpixels have a limited lifespan measured around 5 years (it is getting better). Having a subpixel on for max brightness causes that subpixel to wear out faster, in addition keeping the brightness at maxium generates more heat and the screen can over heat.
IPS LCD has better colors/whites. People are going to try to argue this point for you can make AMOLED have very accurate colors, but in reality though all AMOLED currently in phones/tablets doesn't always keep the color constant for fear of damaging the subpixels.
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They are both good technologies but there is a trade off between each one. I currently have a SG4 which is AMOLED, while my computer monitor is a 2560x1440 ips display. I love both of them.
IPS LCD though is a mature technology and won't have much improvements in the future, AMOLED is a relatively brand new technology which will only get better as time goes by. In the end we are going to see more of AMOLED for AMOLED will allow you to make thinner devices (no need for blacklight), the display can be flexible, faster response times, the subpixels will improve and last longer, and cheaper prices (the process to make OLED panels is similar to an inkjet printer.)