I am going to answer this question with somewhat broad strokes, but realize that it can vary drastically depending on your situation. Fields of study, University of attendance, and other factors should influence how you approach your options.
Generally speaking, a double major will allow you to dive moderately deep in two different subjects. I would think that the difference between a major and a minor would be in terms of course load, which in turn could effect the depth of your learning. That being said, in some cases it may actually make sense to use a minor as a light forcing mechanism and dive deep in your own time, but this requires a lot of self-discipline.
The other benefit would be on paper, however the gains over a minor aren’t that significant. In reality, I think a major and an accompanying minor can be a great option as well. Really depends on what you want to get out of it.