For someone who's pretty well-versed in English, but not a math-oriented computer scientist, this seems like a distinction without a difference. Please remedy my ignorance.
Eatimation is a procedure the generates an estimate, which is a kind of approximation, while approximation is a result value. They are different "types", as a computer scientist would say. An approximation is any value that is justifiably considered to be nearly exact. ("prox" means "near". See also "proximate" and "proxy".)
Estimation is one way to generate an approximation. An estimate is a subtype of an approximation. There are non-estimation ways to generate an approximation. For example, take an exact value and round it to the nearest multiple of 100. That generates an approximation, but does not use estimation.
I’m not sure the linguistic differences here are as cut and dried as you would like them to be. Estimate and approximate are both verbs, so you can derive nouns from them both for the process of doing the thing, and for the thing that results from such a process.
Estimation is the process of estimating. It produces an estimate.
Approximation is the process of approximating. It produces an approximation.
You can also derive adjectives from the verbs as well.
An estimate is an estimated value.
An approximation is an approximate value.
But you’re right that the ‘approximate’ terms make claims about the result - that it is in some way near to the correct value - while the
‘estimate’ derived terms all make a claim about the process that produced the result (ie that it was based on data that is known to be incomplete, uncertain, or approximate)
