The phone battery is bigger. For perspective, an 18650 battery is substantially bigger than a AA, but the best they can do is ~3500mAh. Iirc AA lithium batteries are barely more dense than alkaline or nickel, and are awkwardly high voltage so require a buck corcuit, further harming performance.

Lithium chemistry in a AA form factor, with integrated downconversion and even charging port and circuitry in each individual battery, are cheaply available on Amazon, eBay, newegg etc. Depending on how much you want to spend, they range from no-name companies lying about huge capacities, to real brands advertising (unfortunately in mWh!) genuinely useful capacities, and they are CHEAP. They handily beat Ni-mh batteries in pretty much every way; capacity, self discharge, useful voltage until they die (this actually has a downside in that device battery indicators can't tell that the batteries are running low), cycle lifetime, no annoying management like having to be careful when and how you charge your batteries etc.

High capacity rechargeable NiMH and lithium batteries are 1.2V.