The most obvious thing is in accessing materials which aren't available, at least freely, on online archives. The vast majority of material not made on a web browser is not on the web in any way, shape, or form. One particular source of archiving that I've looked into are legal documents, which are in large part not available for the general public. There's no particular reason for this - they are in the US mostly public record - save for the sheer amount of them that needs to be digitized. If I did not go to the physical archives, a large amount of information would be completely inaccessible to me.
But, there are still reasons to go to physical archives even if you know a large majority of the information. The context of a piece is very important for historians that specialize in physical artifacts. For instance, it's one thing to see the photos from a photo album. It's another thing to see what order and where on the pages these images were placed, giving you a sense of the arranger's artistic sense and how they decided to group collections of photos. These tend to be things that require a lot more analyzing and detective work to pull meaningful detail out of, rather than simply reading a story by itself. What does the medium of an object tell you about the people, time, and place that made it?
There's a wonderment factor to it as well, no doubt. To know that something exists, to possibly handle it with your own hands. That can be revelatory in itself even if it's not specifically a matter of historical intrigue. Knowing that objects exist from hundreds or thousands of years ago is an amazing stoking flame for enterprising historians that want to make an impact. Usually though, any handling of an object is fitted with a greater motive, informational or otherwise.
It completely depends on your field of work to how important it is to get to a physical archive. Archaeologists need to do it a lot more than, say, political historians. There are ways to know how much it can add to a project, but ultimately the historian themselves needs to find value in it to extract that. Ask the right questions and you're halfway there to knowing the answer.
Simply, most of the resources I need aren't digitised. For all the talk of "everything's on the internet these days", it most certainly is not.
If I could get what I needed in digital form, I'd happily use those, but the vast majority of the texts that are of value to me simply don't exist in that form. The things that have been digitised are a lot more mainstream and so a lot less useful to actual research. The things that are really valuable tend to be one of a handful of remaining copies of a text, stored only in some remote private collection.
Online libraries are great for more modern stuff, except even then they can be severely lacking given the dearth of publishers willing to have their published papers be openly available. The number of things I've published myself which I've then been told I'm not allowed to share online is kinda nuts, and that's still the norm in my field. Only a few journals are open access. It's a shame, really.
Publishers are there to make money, not to give access and spread knowledge.