Four days of the 7 Days in Rome tour practically overlap the final 4
days of the 10 day tour. Unless you really like repeating sites so
close to the first time you see them, this doesn't sound like a good
use of time and money. I would say go ahead with the 10 day and then
spend the 7 days in Rome time either continuing to see the sights in
Rome on your own.
That's sort of where i'm at too. I'd take what you learned on your 10-day tour and apply it to exploring Rome independently on your own for a week. Having been there for a short time (2 days) on your escorted tour, you will have an idea of how the city is organized and how to get around, plus have some biggie attractions crossed off the list. Bored? I've spent a total of several weeks in Rome without running out things to see: my must-do list is still very long!
A week will also provide opportunity for a couple of (easy) day trips further afield, like Ostia Antica and Orvieto.
Spend some time with a lovely pile of guidebooks and you won't need a tack-on tour for Rome unless you're wanting the companionship/social piece a tour provides. It's an easy city to see on one's own, and doing so allows you to spend as much - or as little - time doing exactly what you want to on your own schedule.