My question is: are these two states identical? If I were perfect, I would presumably be sinless. But if I were sinless, would that in and of itself be enough to be perfect? Or would I just be "not doing some things" but I would need to change other things in my life to meet my redemptive potential?
The word perfect rarely means sinless in the Bible, it generally means well rounded, or mature. The idea of a sinless Christian is foreign to the Bible, except of course those in heaven.
For example Paul uses the words translated into English as perfect here:
Here paul uses the Greek τελειόω teleioō ; to complete, i.e. (literally) accomplish. It means 'to fulfil', 'to carry out' a required course in the sense of a received commission. As this rare context does actually imply actual sinlessness in that he aims for heaven, Paul, as our example avoids possible confusion saying he has in no way achieved it, but aims for it.
In the case mentioned:
The Greek word for perfect is similar, it is τέλειος teleios. Again it means something similar to complete, it means totality or undivided. It does not mean sinless. From the context immediately before this verse we can understand the logic. God is unrestricted in His goodness, so according the disciples of should be 'total' in their love, even loving their enemies. The totality has to do with loving, not just some, but all. This love would never have been presumed to be a sinless, pure love; just a total love in its well rounded and broad application without favouritism.
You can have a house that's halfway through being built. And the work that's done so far can be flawless, with no problems anywhere, but it's still not a perfect house.
Or, likewise, have a look at Adam and Eve. They were created without sin, but when temptation came along, they chose to transgress God's commandments. Jesus, on the other hand, was also without sin, but he stayed that way even though he was tempted in all points, as we are..
This shows the difference between sinlessness and perfection: perfection implies a state of completeness, in which you no longer commit sin because you've matured beyond the point of yielding to temptation.
Jesus's atonement can take away our sins, making up for past wrongdoings in a way that we are incapable of. But when he commands us to be perfect, that calls for effort on our parts as well, to strive to live the Gospel, to work at improving ourselves until it becomes part of our nature.
Everyone has some sins that they just don't understand. Even if they can see how it might tempt some people, the idea of choosing to do it themselves just feels foreign to them. (Meanwhile, everyone has other sins that just feel natural and normal to them.) My understanding of perfection is that, when our faith and humility, repentance and obedience lead us to the point where every sin feels that foreign to us--which will most likely not be during this life--that that's the point at which the perfection that Christ commanded of us has been reached.