It is inded the screen that flickers. Some people can see the flicker, such as me. Both LED-screens and fluorescent-tube backlights flicker, but at completely different rates. Some fluorescent have a more disturbing refreshrate which is around 300-600 times per second while the more comfortable ones are probably 2000+ per second. But what I've noticed is that those screens that have only one backlight strip of fluorescent tubes in mobile phones are far more comfortable than those that have two. Reason being that the two may flicker at a slightly different rate causing oscillations that can cause nausea.
LEDs can be really good, or way, way way worse than fluorescents. So in that sence fluorescents are safer. This is an article created in 1996 including several testaments from different users: http://www.cloanto.com/users/mcb/19960719lcd.html What you want to concentrate is on flicker, because everything else can be mitigated and tested - and it does not solve the problem! + as per the testaments, there are several who can attest to the fact that it is indeed the flicker. Whether it is LED or fluorescent flicker is of no importance - what matters is the flicker frequency rate and possible pulsating/oscillating light, as with for example flashes of light causing epileptic seizures.
I have owned in the past 5 years approximately 15 different monitors. Some I've sold since they flicker, some I've taken to work. I have five screens here at work, of which two are my main work displays. I've bought the least flicker models (fluorescent) on ebay fearing I won't find them in the future. At home I have a modest two, in addition to my laptop screens.
As for the phone, I don't think iphones are as bad as the ipads - they probably have a slightly different refreshrates. Oled screens ARE better by definition because different colour intensities flicker at a different rate. Thus if you read on an OLED screen against a white background you may feel queazy, but if you choose a light coloured colourful background picture for your background while having black text you will find the reading experience much smoother. Funnily I've noticed that the cheaper the screen, the less flicker. The worst are power efficient Lenovos and Sonys, ipads etc. I suspect that power efficiency and the negative effects of flicker are closely tied together.
I upgraded myself to samsung galaxy note to replace my phone and my ipad all in one.. It flickers very little, still easily perceivable, but I have never had a headache from it. There are definitely better out there, but it's good enough for me and for once I'm happy. A shame really, I love ipad apps.. reading the economist etc. was a breeze. eInk displays are always the best, whether they have wifi or not - airplane mode will not help with this issue. The amount of radiation is miniscule and you would have a fit each time you walk past feet away a wifi-basestation which are far more powerful than what an ipad sends out. Plus if your laptop /w wifi doesn't bother you but ipad does, there's another clue. If you're still wondering whether it's any radiation etc. just turn the ipad backside up. You'll feel instantly better while getting MORE radiation from the antenna which is on the back side of the ipad.
Besides, your health comes first. If you can't use your ipad, don't force yourself. Sell it. Many (like my friend whom I sold mine to) will be happy to use it and will never have problems with it. At least I slept better, without headaches.