Differences between sheaf and bundle

In Mathematics, particularly in Algebraic Geometry, the words bundle and sheaf are used everywhere in the literature to represent different concepts. Cambridge Dictionary defines these two words as follows:

Sheaf: a number of things, especially pieces of paper or plant stems, that are held or tied together.

Bundle: a number of things that have been fastened or are held together.

Furthermore, it gives bundle as a synonym. In my native language (Spanish), there is however a difference between these two words, as one would say that a sheaf is smaller than a bundle. Also, in Spanish, the word for sheaf (gavilla) is used exclusively for stems or branches, such as vine shoots, that is, nobody would say "una gavilla de papeles".

My question is if, despite the synonymity between bundle and sheaf mentioned in the Cambridge Dictionary, there exist more subtle differences between these two words.

Solution 1:

The words are used in a technical sense in mathematics. Their usage in that context does not need to, and in fact does not, bear any relation to their usage in other contexts. The Cambridge Dictionary is unlikely to be the prime source for the technical meaning of such words in mathematics.

As an academic in the mathematical world, I wish that my colleagues would cease to steal normal words and give them technical meanings unrelated to their normal meanings, but nothing I say will ever persuade them to stop.