It depends on what you are looking for in the recording.
For obvious mistakes, timing errors, the iphone is good enough.
Regarding dynamics, it depends on whether you can turn OFF the auto-level of the recorder. If on, it will flatten out all yourdynamics
For the fine nuances in sound, like a bit of harshness etc, this requires really good mics. Some mics may make the sound a tiny bit harsher or less harsh. If that is what you try to get right, better have top mics.
First thing for such a top mic imho is flat frequency response. This because non-flat response will change the harshness in the sound.
I'm using DPA 4060 because they have the flattest frequency response in that price range. Earthworks might be slightly better but much more expensive.
Neumanns generally are not very flat, normally they roll in slowly until about 100-200Hz, and some bumps around 8-15kHz. I think these aim more for voice and speech. I expect them to make the sound slightly harsher and the low basses to be a bit muted.
Now that' s of course great if you are looking for that kind of sound or your piano has the opposite. But not if you are trying to really hear what sound you are making.