Yellowstone/GT area is too large to stay in one spot for 2 weeks.
You got to move around, stay in different areas for a few days at a time.

Mid september might get problematic though. Most lodging and other
facilities (as well as the first roads) close for the winter by the 2nd week
of september so your choices will be severely limited.

If you can pull the trip forward by a few weeks to the last week of august
and the first week of september.
Still good autumn weather, chance or rain or snow (especially high up) but
not that severe.

It will also be relatively quiet, as it's after the main tourist season.

http://www.usefilm.com/staff/duckman for some of what I shot there first
week of september last year.
We stayed at Jackson Lake Lodge and Canyon Lodge (both are central in the
parks).
Not the cheapest maybe but saves you hours of travel (and thus gas).
Had we stayed longer in Yellowstone we'd likely have taken up lodging in 2
locations for a few days each time to save even more travel time.

Facilities are complete but somewhat basic (especially by the end of the
season, expect some things to be closed already).
Don't expect telephone or television in your cabins (those are available in
common rooms). Hot running water is available though (from a boiler, so you
may have to schedule your baths).
Cellphone coverage is nonexistent (certainly my triple band GSM phone got
its last peep in Jackson village south of Grand Teton and then only came
back alive about a hundred miles northwest of Yellowstone a week later).

Bring some warm clothes, dress in layers.
You will get freezing temperatures at night, while daytime can be well over
20 Celcius.

Bring all the film and other supplies you need (reckon about 2 rolls of film
a day and round up, for 2 weeks that would be 30-35 rolls).
There are few camerastores and those that do exist have steeply inflated
prices (especially for the good stuff like Velvia which I was able to get
from one store in West Yellowstone).

Take a monopod and/or tripod as you'll want to shoot at (pre)dawn and around
dusk for the best light on the mountains and the abundant wildlife.
Flash might come in handy for some macro work but I didn't use any (I abhor
people using flash on wildlife).

Jeroen Wenting
jwenting at hornet dot demon dot nl

Regarding the other poster, I have worked in Yellowstone in
mid-September and found it wonderful: far less crowds (in fact no
crowds), no bear jams, and didn't find any facilities closed.
I stayed in Gardner the times I've been in the park in
September.  The biggest problem is that days are shorter,
so less daylight to do things.
If you are deep in the park, you could get back to
town rather late and have fewer choices for dinner.
If touring and not staying in the park, I would split
it between Gardner, West Yellowstone, and the Tetons
Jackson if you can't get in the park).

In the Tetons, try Signal Mountain Lodge.  The lake front
cabins have a kitchen and 2 rooms (1 bath, 1 bedroom and
one larger kitchen + living room that converts to sleeping).
They have other cabins too (smaller).  Jackson Lake Lodge
also has cabins that are cheaper, some (more like hotel rooms)
have kitchens, though I've only stayed in rooms that were
for 2.

Mid September could be peak fall colors in the Tetons.
Yellowstone does not have much in aspens, or other trees
that turn yellow/orange like in the Tetons.

In Yellowstone, the elk will be abundant and doing their
fall "thing" like you've seen in RMNP.  (Same with the
Tetons, but more elk in Yellowstone.)

Call the park service in Yellowstone and ask when services shut down.
At that time of year, there is always the possibility for snow
and road closures, although I've never experienced any.

In the Tetons, it could be busy as many go to see the fall
colors then.

Roger