What are some of the benefits of understanding differential geometry? This question isn't as well-defined as I would like, but here goes.
I'm interested in gauge theory, e.g., Aharonov-Bohm effect, the Berry phase, etc, and I have seen that this can be formulated rigorously in the context of holonomy in differential geometry. However, is there any substantial benefit in formulating it this way, other than it being rigorous? For example, is there some concept/effect that is very difficult to understand unless I know the details of differential geometry. Or maybe is there something that becomes trivial once I understand differential geometry? Examples would be greatly appreciated.
I understand the non-rigorous approach to explaining such phenomena and have some background in (real) manifold theory, but since a lot of gauge theory is formulated in terms of principal fibre bundles and complex manifolds with hermitian metrics, it's quite a learning curve and I'm not sure if it's worth delving into full rigor.
 A: I like your question a lot, and this is isn't an answer but it made me think of something Douglas Stanford said on the awesome podcast "the universe speaks in numbers". (timestamp around 23 minutes)
Stanford was working on a problem and was able to compute some complicated result - he believed the answer, however he had no satisfying explanation as to why he got the answer that he did. He then encountered Ed Witten and after chatting about his result, Witten explained to him how to understand the problem using "fancy mathematics" (what mathematics this means is not specified in the podcast, but knowing who Witten is, it probably was a geometrical insight).
What Stanford said about this is interesting and amusing: he said this was "some simple problem that he cared about" and he thought this should have nothing to do with any fancy mathematics naively. But Witten showed him that day that the fancy mathematics provided the "right" way to derive his result.
I've always wondered what that problem was, and how the fancy math (probably geometry) simplified that Standford's calculation. And your question made me think of that, so here's some food for thought. Again, this is not an answer but maybe someone reading this may know about the specifics of the above problem and this can prompt an answer.
A: This is a classic tradeoff in research-level physics. A more sophisticated approach can often
(a) explain otherwise mysterious results,
(b) provide an invariant formulation that does not depend on arbitrary choices of coordinates, and
(c) suggest new directions and connections that are not obvious.
If you are a student, then learning "standard" tools in your field is also important, even if there is some up front cost, because ultimately you will need to have a new idea and so you have to know the "standard" ideas people in your field will know to try. But, it is also often not strictly necessary to use sophisticated new tools. And it is also certainly possible to invest in learning a new tool, and then find it does not give you a good return for your (time and effort) investment. I also personally have the opinion -- probably not shared by everyone -- that differential geometry applied to gauge theories helps elucidate mathematical aspects of Yang-Mills theory, but doesn't add physical insight (I also have the same opinion about GR, that differential geometry gives us a nice mathematical understanding but not a physical one -- this is a viewpoint advocated in Weinberg's textbook, but again is not a universally held opinion). Of course you have to learn differential geometry to do GR, and parts of research involving gauge theory, but you should just know why you are learning it.
Anyway, the point of this rant is that I'm not going to say whether it is or is not worth learning differential geometry, because I don't know your field, your level, or background, or your goals. On top of that, there is a wide range of what one might mean by differential geometry, from "hardcore" (fiber bundles, holonomy groups) that is fairly specialized, to "vanilla" differential geometry (connections, Lie algebra, differential forms) which basically every working field theorist needs to know. (And that's not even getting into "very hardcore" topics like twistor theory). But I can give an example of (a), (b), and (c):
(a) Differential geometry can explain results in gauge theory. A magnetic monopole can exist because it's not always possible to find a globally defined smooth vector potential on the surface of a sphere.
(b) Berry's phase can be understood in an invariant way as a geometric phase which does not depend on a specific path. The generality is very clear from the geometric formulation, but it might be less obvious if one only looks at specific examples that have to be calculated on a case-by-case basis.
(c) Differential geometry can suggest new directions. Instantons are a major topic in modern gauge theory, and a lot of the language needed to classify and understand these solutions is based on differential geometry.
In the comments, @ChiralAnamoly mentioned a beautiful application to chiral anomalies that is too nice not to bring up. One of the most remarkable results in mathematical physics in the second half of the 20th century is the Atiyah-Singer index theorem, which relates the global topology of a manifold to its local geometry, making essential use of the Dirac operator defined on this manifold. Besides its mathematical interest, this result can be used to give a very elegant derivation of the chiral anomaly, which is a result showing that a chiral symmetry present in the classical Lagrangian is broken at the quantum level. Again, this result can be derived using "standard" methods by computing certain 1-loop diagrams, but differential geometry provides a deeper understanding of what is going on.
