I'm currently running Lost Mine of Phandelver as a new DM, with a group that's about half experienced players and half newcomers. I have easy access to large-format printers since I work as a digital printing operator, so I asked my boss if I could print out a 1.5cm grid (my preferred square size) on a 60x40cm sheet of PVC, laminated with dry-erase film. I worked the image into a nice soft parchment-y look. This gives me a basic blank grid I can reuse at will with markers of various colors. Useful for drawing maps on-the-fly.
Other than that, I have some transparent PVC sheets that I can also use with dry-erase markers, so that I can keep a complete map behind my screen (printed at home with the same square scale as my base grid), take the sheet and overlay it on top and just trace the map over according to what my players' characters can see. Granted, my redrawn map is not as beautiful as printouts from the campaign book, but it's a great way to quickly add stuff by borrowing the map a few seconds and handing it back. it overlays nicely on my base grid.
The only drawback is that any tokens or pawns placed on the board to represent characters and objects will need to be removed and put back in place every time I need to snatch the map, which typically depends on how quickly they advance through the dungeons. It's worked well so far, though, and my players seem to enjoy the exploration aspect and the decent level of preparation involved. having correctly proportioned maps thanks to the grid helps a ton with movements and ranges.
Crudely drawn maps are great if you can voice elaborate descriptions to fill them with textures, sounds, and smells they can relate to; the original high-definition illustrated map isn't really necessary for the players if you're doing your job as DM well enough.