Passing tones move by major or minor seconds between two chord tones that are a third apart.
According to Russo, a neighbor tone is a non-chord tone that is a major or minor 2nd above or below a chord tone, preceded by a skip or a rest. According to other sources, a neighbor tone occurs when you step up or down from the chord tone, and then move back to that same chord tone.
According to several sources on the interwebs, auxiliary tones are essentially neighboring tones. For Russo's purposes, aux tones are non-chord tones that move away from a chord tone by a major or minor second and return to that same chord tone - no preceding rest or skip necessary.
Anticipation tones belong to the chord that follows and are repeated when the following chord "arrives".