Safer? Yes. Safe? No! Think for a minute and ask yourself "Why does the NEC require a sticker that says 'No Equipment Ground' if the receptacle is safe?".
The key to this answer is your question asks about "appliances that expect a ground".
The sticker is there to warn you to not use the receptacle where a ground is required.
Here is the edited NEC section that allows GFCI protection for 3 wire receptacle replacement for ungrounded receptacles, note most carefully the Informational Notes at the end.
406.4(D)(2) Non–Grounding-Type Receptacles. Where attachment to an equipment grounding conductor does not exist in the receptacle
enclosure, the installation shall comply with (D)(2)(a), (D)(2)(b), or
(b) A non–grounding-type receptacle(s) shall be permitted to be
replaced with a ground-fault circuit interrupter-type of
receptacle(s). These receptacles or their cover plates shall be marked
“No Equipment Ground.” An equipment grounding conductor shall not be
connected from the ground-fault circuit-interrupter-type receptacle to
any outlet supplied from the ground-fault circuit-interrupter
Informational Note No. 1: Some equipment or appliance manufacturers
require that the branch circuit to the equipment or appliance includes
an equipment grounding conductor.
Informational Note No. 2: See 250.114 for a list of a cord-and-plug-
connected equipment or appliances that require an equipment grounding
Informational notes remind you to refer to other codes or parts of this code that have bearing. Note 1 generally refers to NEC 110.3(B).
110.3(B) Installation and Use. Listed or labeled equipment shall be installed and used in accordance with any instructions included in the
listing or labeling.
So if you have a window AC unit, and the user instructions (which are part of the UL Listing) or tag on the cord says to use a grounded outlet then it is not certified for use on an ungrounded circuit, and use of a labelled receptacle does not change that.
NEC 250.114 goes further explaining certain equipment, labelled or not, is not safe enough without a ground.
250.114 Equipment Connected by Cord and Plug. Exposed, normally non-current-carrying metal parts of cord-and-plug-connected equipment
shall be connected to the equipment grounding conductor under any of
the following conditions:
(3)In residential occupancies:
(a)Refrigerators, freezers, and air conditioners
clothes-drying, and dish-washing machines; ranges; kitchen waste
disposers; information technology equipment; sump pumps; and
electrical aquarium equipment
(c)Hand-held motor-operated tools, stationary and fixed motor-operated
tools, and light industrial motor-operated tools
(d)Motor-operated appliances of the following types: hedge clippers,
lawn mowers, snow blowers, and wet scrubbers
(e)Portable handlamps and portable luminaires
So an ungrounded GFCI receptacle will provide protection in case of fault to ground with a two-wire circuit, and allows polarized plugs to replace old non-polarized plugs, but it is not legal to use those receptacles for many applications, which is the main reason you are required to label it "No Equipment Ground".
And you asked difference about grounded or ungrounded GFCI's, at least one aspect is if a fault develops to the ungrounded chassis the chassis will remain energized until something or somebody completes the path to ground. A GFCI functions by running both wires through a current sensor (current transformer). If no current is leaking to ground the magnetic fields cancel each other and nothing is detected in the current transformer. If there is an imbalance then magnetic fields don't cancel, current is developed in the CT, and the device is triggered to open. So just energizing the chassis doesn't upset the current flow, a grounded chassis provides that path. I think most of the items listed in 250.114 are there because human contact with the chassis is the likely event to complete the path.