DOD Approved Changes to Campaign, Expeditionary, and Service Medals

No. You either get a campaign medal or the expeditionary medal. Not both. So you got the campaign medal and are wearing that. If you wanted to do the paperwork, you could get the GWOTEM and trade in your campaign medal.

Before Afghanistan was a named campaign, they gave us the GWOTEM. After they gave it it's own medal, they said we could choose up until 2007(ish) but not wear both. Since GWOTEM applies to regions other than Iraq and Afghanistan, the country campaign medal is more prestigious in my opinion.

I suppose, if you went back to OEF, you could argue for the GOWOTEM instead of putting additional campaign stars on your ribbon, but not sure why you'd want to.

GWOTEM is awarded for service toward the GWOT while deployed to deployed locations, mentioned in the article, supporting the Operations in the article. GWOTSM is awarded for support of the same operations while note deployed. For example, some of us were sent to CENTCOM in Tampa, where we supported those operations, but we did not deploy into the CENTCOM AOR. As long as your dates don't conflict, yes, you can wear both. But someone who is sent to Tampa and forward deploys to the AOR, would only get one (I'd take the GWOTEM). The Service medal is really easy to get. While stateside (or in a location not mentioned in the GWOTEM criteria) if your NCOER says anything "in support of" the named operations, you're eligible.

I have the GWOTEM for deploying in 2002 to Saudi/Afghanistan to support OEF, and the GWOTSM for "deploying" to Tampa to support OIF/OEF (where I did not get to go forward). Others have received the GWOTSM for say loading bombs in CONUS for a strike mission which supported one of the named operations.

Added Edit: Campaign Medal adds a little more complexity. As another poster stated, originally there wasn't a campaign medal for Afghanistan. Later they added it and said you can trade the GWTOEM in for it. I chose to keep my Exp medal. In 2013 I deployed again to Afghanistan and got the Campaign medal. Personally I think that particular rule was dumb. If one deployed to Afg, they should get both the Afg Campaign Medal and the GWOTEM. If they went to Iraq, they should instead get the Iraq Campaign Medal and the GWOTEM.

During my first deployment, in Iraq, we were awarded the GWOTSM. I was awarded the GWOTEM for my recent deployment to Kuwait. If you have received a memo in your IPerms stating that you were awarded it, you can wear it. If you are lacking the paperwork, speculation isn't something you'll want to take as 100% accurate. Until you have the supporting documentation, just wear the GWOTSM.