I much prefer to use physical books at the table, but running Dungeon of the Mad Mage for my group is proving a bit complex and I'm wondering if using the DndBeyond digital version on a laptop would work better.

However, I'd rather not buy the book again without knowing exactly what I'm getting...and nothing I've found seems to be very clear on that. Things just say "access to the book in digital format".

Is it a hyperlinked series of pages? A view-only PDF? Or what?

Series of hyperlinked pages. If you already own the physical version, I don't think it's worth it to buy the digital one where adventure modules are concerned. Maybe for the maps if you buy it on Roll20 (though that seems pricy just to get convenient digital maps), but not here.

To get a feel for what you get as “access to the book in digital format,” take a look at the free Frozen Sick adventure. 

If you bought the full “book” here (rather than the compendium only option), you would also get access to any monsters printed in the books in both the Game Rules>Monsters listings and in the encounter builder, along with magic items, etc., although there don’t appear to be magic items, spells or character options in Dungeon of the Mad Mage.  Note that even with the compendium only purchase, you would be able to see monster statblocks when viewing the “book” here (from the Sources menu)

As for whether or not it would simplify running the adventure, that depends.  When I ran TOA, I found it helpful to have access to both a physical copy and the version here, although I only paid for the latter. (I was running the game at the library and the library paid for the physical book). I found the extensive hyperlinking helpful, and the fact that i could have several different pages of the book open at once by opening them in different tabs.  

That said, what works well for one DM doesn’t necessarily work well for a different DM, and you are the only one qualified to make the cost/benefit analysis for yourself. 

The advantages of having a book on DDB as well as physical are two fold, in my opinion:

1. The minor one is that you get access to the character generator with the options.
2. The more important one is that you get links in the adventure to statblocks. If you follow along the adventure as DM and the party comes across, say, an Owlbear that attacks them, you can just click on the name of the Owlbear in the text and it will bring up its statblock for you to use in the encounter. You don't have to go rummaging through appendices or other books - it's right there.

Personally, I prefer physical books. I use cards and they're much easier again to run an encounter with (if you have two different enemies, say a bunch of Zombies and a bunch of Anchorites like I did a while back, you have to keep going back to the adventure, selecting the other creature and so forth, which makes things clumsy and slow with DDB, books are slightly easier because you can use bookmarks, but cards have all the info right there for you), but that functionality was really useful. Having modified maps that players can see (without giving away secrets) is a nice bonus as well. 

I strongly prefer the digital tools available on DDB, but everyone is different. My recommendation is to run a one-shot using just the basic rules with the free adventures available here. If you like the way that works, the paid books will be basically more of the same.