Anthony here, Mogreet CTO. MMS is not at all like SMS under the hood. While links are theoretically supported, in practical real life the content (images, video, audio, etc) is encoded into the MM7 body, and delivered all together to the handset. Check out our docs online for much more about what you can do:
https://developer.mogreet.com/docs
I can't speak to the Twilio API, but you certainly can with the Mogreet SMS/MMS api. When an MO (Mobile originated) message is received from your user, we decode and store any media included (like images!) and then we make a web services callback to your app with the message content, the info about the user that sent it in, and a url to the media payload of that MO message.
Anthor