There is a Military Disney Website that goes into this in detail. The BIG difference is this:
When you order via Shades of Green, the tickets will be activated already. Which means you can go straight into the parks. They will require lots of personal information via email (SSN / Retirement Date/Etc). They will then call you to get your full credit card number. Then, the tickets will be sent via Fed Ex and this must be signed for in person. (You can have it routed to a pick up location, but the name on the reservation must match your ID)
Despite this hassle, we order ours via Shades of Green.
When you use the MWR sites, the tickets are not activated and you will need to visit guest services on your first park day. Also, if you are traveling with a family members that are not military or retired military, you must be present to activate ALL the passes. This can slow you down in the mornings AND if you have folks that are wanting to go to the parks later or at different times, the active military member/retired member must be with that person to activate the pass.