There are some good threads on this forum if you do a search. It can be a long discussion.
Part of the problem is that none of the ply based failure criteria work well for notched laminates, damaged (impacts) laminate, or fastened joints (either original installation or for repair). Most structure will be designed for those conditions as opposed to unnotched (pristine condition). So right from the start, there are some major challenges.
Max strain/Tsai-Wu/etc. do a "reasonably" well job at predicting unnotched strength (certainly better than the above listed scenarios), provided the laminate is designed "properly". But again, that is of limited benefit since structures aren't designed for pristine conditions. Some/most argue that even for this simple case that the ply based criteria are insufficient (based on extensive testing and comparison to various criteria). Of course, the challenge is that laminate based testing is expensive and hence there is a tendency to try to use ply based criteria.
But long story short, the Tsai-Wu criterion is ply based and does not work as well "in-situ" (as part of a laminate). This is a challenge with any ply based criterion. Even if you had a perfect ply based criterion, it does mean it works at the laminate level. In fact, ply based criterion have no consideration for interlaminar stresses (which can affect unnothced strength and will certainly affect notced strength). During that era, there was hope that ply based criteria would work well for laminates and then notches, etc. They were just proposed criteria (of which there were many), but they did not work as well as hoped for either unnotched or notched strength prediction. It is just a challenge the industry faces when it comes to composite failure criteria. Semi-empirical approaches can be used in lieu of a robust failure criterion.
But to more directly answer your question, the advantage the max strain approach is that it you can decouple the failure modes (fiber and matrix). Some use this in various ways (such as to discount matrix failure modes). The interactive ply based criterion (such as Tsai-Wu) are not capable of this and the theoretical advantage of the interactively is not realized. In fact, if the matrix failure modes are subcritical in a laminate (which we know they can be) then any interactive ply based criterion will lead you astray. So in the end, non interactive criteria are preferred over interactive criteria (but any ply based criteria is still insufficient for the practical cases of notches, etc.)
Brian