The Advanced GM screen contains information from the Game-master’s guide that the main GM screen doesn’t have.
Correct (a list is below).
The contents of the GM screens is as listed on the GM screen section section linked above.
Correct, and then some. The AoN screen has things that aren't on either of the physical ones. (A list is below)
Does the Advanced GM screen omit any information that is present in the basic GM screen?
- Basic/Special Action list
- Skill Action List (lists actions each skill has and what level of proficiency is required)
- Hero Point usage
- Specific Skill DCs (such as the DC for identifying a magic item or recall knowledge)
- Turn Order and Icon Key
- Exploration Activities
How do the two advanced GM screens differ?
AFAIK content wise they are the same, but I don't own the portrait version so I can't validate that.
Exactly what AoN GM screen rules are omitted in which screens?
What's missing from each screen is in one of those lists already. The following things are on neither screen but are on AoN:
- Earn Income table
- Temperature table
- Weapon traits table (what each trait does)
Are there any other differences that define the sort of GM each screen is suited for?
The Advanced GM screen includes rules from the Gamemastery guide that the standard screen does not:
- Creature Numbers (for creating monsters)
- Hazard Numbers (for creating hazards)
- Victory Points (a subsystem)
- Reputation Points (another subsystem)
If you are using these systems on the fly, this will be very useful. If you are not, they are not useful at all. In the case of Creature and Hazard numbers, on the fly means "you are making up a creature or hazard in the middle of a session". If you are making a creature in your prep, this information doesn't need to be on the DM screen as you will have its final numbers already.
The standard GM screen covers things that are always useful, and especially for a newer DM are things you will be looking up frequently. Some of them get easier to remember later and if the list above of what's removed from the Advanced screen is things you already know anyway, then the Advanced screen will probably serve you well.
But if you are looking those things up frequently, the standard screen is going to be used more often as a reference.