First, an observation: Here are two examples of when different notions of weak equivalence in a category turn out to be the "same":


* (Whitehead's Theorem) Two CW complexes are homotopy equivalent if and only if they are weakly homotopy equivalent (i.e. there is a map between them which induces an isomorphism on homotopy groups).
* (An 'easy exercise') Two chain complexes of vector spaces are chain homotopy equivalent if and only if they are quasi-isomorphic (i.e. there is a map between them which induces an isomorphism on homology).


In both cases, one direction is immediate, the other direction takes more work. This prompts my questions:


1. **What are other examples of results that share this flavor? i.e. "Two objects are [weakly equivalent in way #1] if and only if they are [weakly equivalent in way #2]."**


and


2. **Is there a result in model category theory which says "Under such-and-such hypotheses, two notions of weak equivalences are the same" so that we can see the two examples above as corollaries?**


*(I'm completely new to model categories, so apologies for any/all naivety.)*



When are two kinds of weak equivalence 'the same'?
The notions of weak equivalence are not 'the same' in your examples, rather they coincide on a certain class of objects, namely those that are 'cofibrant-fibrant' as mentioned in Arun's comment. Perhaps there is a more general question hidden in yours, however, as in both examples you have one notion of homotopy equivalence based on a cylinder or cocylinder functor and therefore feeling somewhat geometric, and one notion of 'quasi-isomorphism' relative to some functor, that I will write as $F$, to another context / category, $f:A\to B$ being a 'quasi-isomorphism' if $F(f)$ is an isomorphism. This is then a slightly more general question than yours in the context of model categories and is explored in Baues' work on algebraic homotopy. 


