Mobile Signals and Amplitude Modulation
Also, if newer techniques save more power, why is it still used for radio transmissions?

Legacy. You can't just re-equip all airplanes with new emergency radios, it costs a lot.
That's about the only place where AM is still in used not being currently replaced.
For example, audio AM broadcasting is definitely on the way out. Europe doesn't have any medium wave or short wave AM stations anymore, and the long wave stations are mostly for &quot;political communications&quot; purposes (maybe a couple thousand people listen to music on AM – on ships in the middle of the ocean).
Even in large countries with low FM broadcasting coverage (thinking of India, mostly, but also of Australia), it's currently being phased out. DRM (digital radio mondial) is its replacement. India, for example, has installed really high power transmitters for that standard, clearly planning to shut down the AM stations over the next couple of decades. It just takes time – there's millions and millions of listeners who depend on AM radios.

[…] for communication between mobile cell phones and why is AM not used? Also, if newer techniques save more power, why is it still used for radio transmissions?

AM is an analog modulation technique. All cell phones are digital devices. That means AM is not even an option.
Generally, AM has pretty bad quality that you get for a given product of SNR and bandwidth, when you compare it to digitally encoding the speech, and then using an appropriately robust digital transmission system. So, you would need much more spectrum and transmitter power in each cell phone and each base station to even be able to do the same amount of calls.
This isn't new – it was realized in the 1980s, and all cellular voice communication standards that have been designed since then (ca 1985) are digital, because AM sucks just so much in terms of spectral efficiency.

What modulation technique is used [in cellular voice communications]

That depends on the standard. GSM (2G in most places) uses GMSK as the physical layer modulation, and the speech is encoded using G.711, or the &quot;GSM codec&quot; (in two different modes), depending on the available signal and bandwidth.
Everything after that is so flexible that there's no single answer to your question, but complex standards and complex audio codecs.