GTC, it matters because he's curious. So usually when the deacon goes through the open royal doors he is doing something... ie, carrying Holy things, like the Gospel or the diskos during the great entrance. When he's going out to incense he goes through the deacon's door and comes back in the other deacon's door.
And the litanies he goes out the deacon's door and comes back in, I can't remember but I don't think he goes out the royal doors for any litany, but I could be wrong, it's early/late and I need coffee to think, plus there are a lot of services to think through and I can't remember all the rubrics for them, I hope I helped.
UN! has got it, mostly. If the deacon's carrying something, he goes through the royal doors. If not, he uses the deacon's doors.
A breakdown of what I do:
When I do the litanies in services, I use the deacon's doors.
At the little Entrance, I go through the Royal Doors with the gospel.
When I go to do an open-door lesser censing at the epistle and the cherubic hymn, I go through the royal doors.
When I take the gospel out and return from reading it, I use the royal doors.
At the Great Entrance, obviously I go through the royal doors with the paten.
And, in several places in the OCA, deacons are blessed to distribute communion if father is the only priest and needs someone to help with communion due to numbers. This is the case with where I'm currently serving during this, my last year of SVS, so I bring out the chalice ("In fear of God, and with faith and love, draw near") and when I bring the chalice back in.
The one, quick exception to the "deacons can go through the royal doors if they're carrying a holy object" rule is this: At the end of the troparion/kontakion sequence following "Come Let Us Worship," I get a blessing and "end around" through the royal doors to go to the front of the icon of Christ, finish father's sentence ("For holy art thou, O our God, and unto thee we ascribe glory, to the Fr, the Son, and the HS, now and ever...") and as I finish w/"and unto ages of ages", I rotate around to the people and go back through the royal doors.
Ultimately, it's a difference of practice (I was surprised by your OP, Basil, as I can't conceive of how a deacon could get through a liturgy easily w/out using the royal doors), but that's how I've been trained to do it.
Wow; been a while since I've posted here.
Hope it helps!