>Is it just because one involves repeated payments?
No, it's illegal because it involves reselling streamed or broadcast content (as opposed to content on physical media which can be resold under the doctrine of first sale) which is illegal regardless of whether it's done for one-time or ongoing payment. Commercial re-transmission is the real bone of contention.
>Would it be legal for Aereo to charge a one-time fee instead of a subscription service? Or is the problem that Aereo owns the property where the physical antenna resides? Does my personal antenna+streamer have to reside on property which I own directly?
Probably irrelevant, since they're still running a commercial service. That means the signal in question is not being pulled down by a private individual for personal, non-commercial use. It's being pulled down by a commercial company for the benefit of its customers. That puts it on the wrong side of the law.
Of course, if Aero were set up as a co-op, and its customers were owners of not just the antennas but the building etc. and were mutually responsible for the electricity and so on, then perhaps it could get around the law. But that's purely hypothetical, where as the Supreme Court was ruling on the (very different) reality.
>What if I rent my apartment? Heck, my Internet connection is a subscription service, and I neither own nor operate any of the Internet equipment myself.
If you rent your apartment from a landlord who charges you an additional fee for access to the wire running from the building's rooftop antenna to your flat, then the landlord is the one on the wrong side of the law.
As far as your ISP is concerned, they are treated by law (as far as content goes) in the same way that the telephone companies are treated. That is to say, they cannot be held partially responsible for any infringing uses you may use of their service, in the same way that the phone company can't also be charged with conspiracy when two gangsters use their system to plot a crime.
In terms of the internet, this comes from the Safe Harbor provision of the Digital Millennium Copyright Act (DMCA, 1998. In determining whether a network operator is partially responsible for infringing uses, the critical test is the source of the content. If they were the ones who put it on the network, they they can be held liable. But if (as in the case of YouTube or Dropbox) they're simply the hosts for material that others upload at their discretion, they are protected from liability. There's major caveat in that they must pull the content down if the owner of the content notifies them that its presence on the network is infringing. If they fail or refuse to honor a take down request, then they lose their legal immunity.
Aero is not protected by Safe Harbor since they're the ones who introduce the content to their network when they initially pull it off the air. Again, if they were run as a co-op (meaning there was no legal difference between the owners of the equipment and its end users) the case may have gone differently. But as they are an unprotected third party placing themselves in the middle of a legally protected broadcast transmission, and doing so on a commercial basis, they're doing so in violation of the law.