CDIs must be certified with an agency and are used for official business like courts, government agencies as well as doctor offices.
DIs are ASL screened on a regular basis and they are just the same as CDIs. DIs are more for the relaxed scene like classrooms, social scenes and so on.
CDIs dress very professional, DIs do dress professional and they are encouraged to dress casual so they are comfortable during social scenes.
I have a DI in my classes. He is wonderful because I miss the context during classroom discussions, everyone is Deaf [some is hearing and HoH]. When a discussion starts, I miss everyone's input, this is where the DI is handy. I have a severe neck & back problem right now because for 6 weeks I had to move my head left and right to see who was talking. It all catches up so I'm struggling with migraines on a daily basis.
My DI arrived yesterday and oh I was like a little girl on Christmas Day!!

He made me happy because he mirrored everything everyone said. The prof and the students... I didn't miss anything. Their facial expressions, even the tongue sticking out, he does it. Oh joy! Nothing is missed.
This is why I really advocate the use of DIs. They are not just for DeafBlind people, they are for Deaf people who are not confident with the use of the interpreter alone. DIs do facilitate the interpreters. Deaf people from other countries do feel awkward with the interpreters so they feel at ease with the DIs because DIs do well with facilitating with Universal Sign Language.
It is very important to have DIs and the Deaf Community be involved by training to be a DI as well.
During the International Deaf events, you will see a lot of DIs, they are very important to everyone. They provide facilities such as interpreting between two people. They translate between two languages, a go-between two countries so to speak. So important to have the DIs, I cannot emphasize it much more.
Hope this helps!! HUGS!