I have a couple of Rode mics - NTG2 and NT3

Is there any difference/advantage operating these mics by either battery or phantom power?

I understand they - at least the NT3 - can use 12v phantom power. Is there any difference/advantage operating them in 12v phantom verses 48v phantom power?

Its mainly for flexibility. It can be used with a camera or device that doesn't supply phantom power. Believe me that is handy if you ever get caught in a situation where you can't provide phantom power. The convenience comes at a price, forget the battery in there and put it in storage it can corrode and ruin the mic, or forget to put a new battery in and it dies in a middle of a performance, or slowly weakens over time and you don't realize its the battery. Battery also adds more weight and size that is noticeable if your using it on a hand held boom. Batteries provide very clean power but I can't say I've noticed the difference between phantom power.

I have not looked the RODE specs, but in general, Phantom/Battery powered mic's SPL range and S/R noise would be pretty close. As they are 'Electret condenser microphones', a fixed-charge plate mic which does not require a full 48 volts of the 'true' condenser'. Normally just requiring low-voltage 'Bias current',.. either supplied by a reduced Phantom power or a 1.5 or 9v battery.

You will typically get a bit more dynamic range on phantom, compared to battery. Most mic vendors show different specs for the two situations.

Plus running on phantom power versus battery ever so slightly reduces your ongoing costs and non-recyclable garbage footprint.