I was thinking about starting a head-to-head comparison thread, since I have both, but I'm just as happy to chime in here.
DMG II and Advanced Gamemaster's Guide are both good books. There is a good deal of overlap. AGG, I think, is aimed at slightly more experienced DMs. DMG II really is an extension of the core DMG, in many ways; AGG has some of that content, but also touches on some of the variant types of rules like Unearthed Arcana.
DMG II begins with a discussion of Gamer personalities, and the sorts of issues that can come up because of them. So, too, does the AGG. Both offer advice on table rules and how a DM/GM can better present information. These sections contain advice applicable to any game system, even if sometimes phrased as D&D examples. Of the two, AGM spends more time on "timesaving tricks at the table" (such as Initiative cards, a photocopyable set of which are provided in the book, and making area templates for area effects), while DMG II spends more space on interaction issues. Both talk about effective presentation, and adding drama.
Both books also address running adventures, and touch on adapting published adventures to your needs.
AGG and DMG II both provide a selection of additional rules.
DMG II has a lot of material on Cities .. handling crowds in them, owning a business in them, the Mob template for when crowds turn violent, the administration of justice, and finally the City of Saltmarsh as a fully-detailed example of putting it all together.
More of AGG's rules are Combat oriented (Bushwhacking, Class Dodge Bonuses, Critical Hits that cause special damage effects, and new conditions like "groggy" {for when you just woke up in that nighttime raid}), but not all of them are. Experience debt, for example, prevents you from losing levels due to XP costs of various types; instead, these amounts become a deficit applied against your future earnings.
AGG talks a bit about both High and Low/No Magic campaigns (a big plus in its favor), the ways standard D&D has to be adjusted to fit those variants, and provides some discussion on the pros and cons of different methods (including the spellcasters-must-evenly-multiclass approach).
AGG also has a few of the little details that made the 1st Ed DMG such a valued resource: tables of "decor". 40 Plot points ... 40 Battle scenes ... "Urban Set Dressing"... "Wilderness Set Dressing".
Both books touch on Mentoring and Apprenticeship. AGG spends all of its time on Mentor rules, and how they affect the PC. DMG II spends equal amounts on Mentors and Apprentices, as well as how a PC can begin as an Apprentice and become a Mentor. Both takes seem pretty good, so it is just a matter of which approach fits better into your campaign. You could also meld them a bit, and use some of the AGG Mentor details with the DMG II setup.
Both books deal with NPCs, but in different ways. The DMG II is more mechanics-focused, giving advice on how to determine the appropriate level, gear, and fees for a hireling so s/he is competent to help the PCs without being a cheap bypass. This leads into some example Complex NPCs, presenting stock NPCs that have 2 or more classes (and/or Prestige Classes). It also discusses how and why to design a Prestige Class, especially offering tips on how to decide if it is balanced. By contrast, the AGG offers 1st-20th level lists of the PC classes from the Advanced Players' Manual, with detailed workups of 5th, 10th, 15th, and 20th levels for 1 of each.
In the end, I think both are useful. DMG II is, as I said, more of an extension to the core DMG. It covers material that every DM/GM will find useful to have around, though experienced DMs/GMs have most likely already come up with their own answers to the same questions -- in that sense, it is more newbie-friendly. AGG does more to address campaigns with alternate levels of magic -- there is a good-sized section on special-though-non-magical materials, for example, that allows DR to play more of a role in low-magic or no-magic campaigns (where the standard DR would otherwise become vastly more powerful).
There is a lot more in each book that I have not touched on. Despite the occasional overlap, I am pleased to have both.