What is the difference between the "Amplification" and "Gain" parameters?

The "Amplification" parameter increases the quiet parts of the audio recording. Amplification occurs during playback and also when merging or editing. In the later case, it modifies the audio data to include the amplification whenever pieces of a recording are merged. Merging happens for almost every memo, which is why this parameter is often 0 for existing memos. If the "Auto normalize" setting is disabled, "Amplification" is ignored.

The "Gain" parameter influences the analogue to digital conversion of the recorded audio. On a modern iPhone, it is only available for external audio sources (not for the internal microphones). The "Gain" parameter should be adjusted so that the volume indicator never reaches its maximum value (when it reaches it, then clipping will occur).

Audio engineering details.

A microphone is a device that converts vibrations in the air (sound) to variations of an electric current. These variations can be represented by a real number. This is an analogue value that changes all the time. But to be represented in a digital equipment (like a smartphone or computer), this ever changing value has to be converted to bits. This has two consequences.

First, even if the value changes all the time, the digital representation can only contain a limited number of changes. This leads to the parameter called "Sample rate", the number of times the value is saved per seconds. The higher the better, but also the more bits it takes to store one minute of audio. The human ear is known (on average) to only be able to hear the difference up to a sample rate of 44100 Hz. It is an important parameter, but is not related to the "Amplification" and "Gain" parameters.

Second, the value has to be represented by a given amount of bits. Every bit multiplies by 2 the amount of different values that can be represented exactly. This is where the "Gain" parameter is important. It should be adjusted so that the sounds being recorded use most of the representable values. The louder the sound, the bigger is the number that needs to be represented. In the most common case, there is a loudness limit above which all sounds are recorded with the same value. When this happens, the recorded sound is distorted (it can be heard during playback and can destroy the quality of the recording). This is called clipping and should be avoided. Reducing the gain helps avoiding this problem, but has to be done before clipping occurs. It is quite hard to removing clipping during post processing of an audio file, so this parameter has to be adjusted before recording.

When listening to a recording, you can find that what you want to hear is too quiet. This is especially the case for voice recordings, where you only want to listen to the voices and other sounds are irrelevant. Audio Memos uses an audio filter called a compressor to amplify soft sounds and reduce loud sounds. The "Amplification" parameter is the gain parameter of a compressor filter. This can be done after recording. Its can even be done during playback. However, there is no standard way of specifying that a compression filter should be used during the playback of an audio file. If audio files were only played inside Audio Memos, this would not be a problem. But as the files can be stored in iCloud Drive or shared, Audio Memos needs to modify the audio in the file to include the result of the compression filter. Users that intend to post process their recordings themselves can disable the "Auto normalize" setting to disable the compressor filter.