I know there are a lot of differences between these two games, and I don't mean to ask for a comprehensive list. What I would like to know is if there are any important differences between the D&D 3.5e Dungeon Master's Guide and the PF GameMastery Guide. I have the impression that these books contain mostly general DM/GM advice and are somewhat light on game-specific rules. Can I GM a PF campaign with just the DMG? Or do I absolutely need the GMG?
You only need the Pathfinder Core Rulebook to run a game. The GMG is not the equivalent of the old DMGs; the core rulebook is basically PHB+DMG. The GMG has help, additional rules, advice, NPCs, etc. You do not need the 3.5e DMG either.
You can see a lot of the content (specifically the rules content) of both the CRB and the GMG on the PRD site to compare and contrast - that omits a lot of the "advice" sections however. The CRB has about 10 pages on running campaigns and 30 on environment (not really worldbuilding in the entire-setting sense). The GMG has a lot more on structuring campaigns and building settings, but that's a weird definition of "information you must have to run a game" because that's more general "read it once for any game, you're good" kind of content.
Pathfinder does not follow the three-book PHB/DMG/MM tradition that D&D does. Although PF's Bestiary series serves the same purpose as D&D's Monster Manuals, PF's Core Rulebook is more like a combination of D&D's PHB and DMG, as opposed to being analogous to the PHB alone. You can certainly use the fluff from 3.5e's DMG if you wish. You can even use some of the rules, as long as you give precedence to PF's rules when the two disagree.
PF's GameMastery Guide is more analogous to what 3.5e called DMG2, which wasn't necessary to run 3.5e campaigns (and wasn't released until several years after the release of 3.5e). As with that book, it can help with some of the various behind-the-scenes DM work, but isn't really necessary.