How do you call it when two celestial bodies come as close to each other as they will in their current orbits? On Earth, we say Mars is in opposition when it is 180º from the Sun, which also marks the times when Mars and Earth come closest to each other.

Is there a specific name for such proximity events? For example, Saturn and Jupiter come to this maximal proximity every 20 years or so. Is there a name for the event itself, the "approaching" (previous) phase and the "departing" (next) phase?
This is a syzygy.  "A syzygy of the Sun, Jupiter and Saturn." In general a syzygy is when three bodies are in a line.  In your case the three bodies are the sun, and the two planets. You can also talk about "Saturn is at opposition as seen from Jupiter. Not every syzygy is "proximity event" but many are.  I know of no word or phrase for the "approaching phase" or the "departing phase". 

I don't think that there is a specific term for "closest approach". Establishing absolute distance is generally difficult, compared with finding the direction. And the closest approach doesn't happen exactly at opposition (due to orbital tilt and eccentricity). Terms like "opposition" or "syzygy" are about the direction rather than distance.