An estimation is the act of estimating, or guessing, about the quantity, quality, or other aspect of an object or person.
An approximation can be an object or person that is almost exactly like something else, but not quite, either by defect or design. If by design, it is not meant to be exact. The adjective form is approximative.
In general, an approximation has more information and comes more intentionally closer to the mark than an estimation.
Side note: A rare, but related, word is proximation. A proximation is the closest to something, something that is nearly correct or is approximate.
“You need to understand everything is based on my estimations,” he says. “All the radar does is back up my estimation, that’s all it does.” [The Daily Telegraph]
Because Sophie represented a unique opportunity to cross-check two methods of weight estimation on the same skeleton, the sum — and methods — can now be considered more accurate. [The Washington Post]
It’s easier to see what she does as an approximation of what she will do. [CBS News]
In the last census of life on earth conducted in 2011, around 8.7m species were calculated using statistical approximations and that excluded bacteria and small organisms. [International Business Times]
The Communists have also failed to find themselves a real leader. No Mexican revolutionary leader unfurled the Communist banner. The nearest proximation, Vicente Lombardo Toledano, came from the intellectual class, not from the military or the workers. [Foreign Affairs]