Well, not being an expert, I'll take a stab at it.
Tape saturation is the point at which the analog tape has stored the maximum amount of magnetism it can hold, without distorting. Beyond this level, clipping will occur, which is a type of compression.
True analog clipping and digital clipping will sound different, by definition. As they say, when analog overloads a little, it sounds 'warm', and when digital overloads a little, it sounds 'harsh'.
I don't think 'tape compression' is a 'straight' technical term, or an analog term, at all. I think it's a term devised by digital recording audio engineers, to describe actual analog tape saturation, and the relative 'natural' , 'warm' clipping sound thereof. I think the digital people coined that term, 'tape compression', when referring to their digital-plugin-software gizmos, that are software algorithms used to get a digital recorder to mimick a 'natural' analog 'warm' clipping sound.
I don't recall any reference to the term "tape compression" in actual analog recording, but I believe it's a term the digital crowd coined, to describe the effect they desired to have on their digital recorder, to... [get this]... make it sound more like analog. Is that ironic, or what?
That's what I know of it, anyway.