I'm pretty sure that the symbol of the red cross has transcended into more than just a corporate logo, and is recognized as a symbol for medic items. I don't think this is a bad thing, nor that they want or can do much about it.
It's not a corporate logo though. It's legally equivalent to a country's flag, in international law anyway. There are individual not-for-profit "companies" allowed to be set up in each country, but they are heavily restricted in their charters.
Aye, it's strange they're so concerned about the usage of their icon as one for life saving medical items, (It's not like it's some kind of bad PR having their symbol being associated with life saving medical supplies) one would think the Red Cross would have higher priorities on their agenda then this petty legal mumbo jumbo.
I think part of the problem was that the symbol was being used by "for-profit" groups (like first-aid kit manufacturers). So they decided to politely remind people that the Geneva convention says that it can only be used either on military medical gear, or on free hospitals (so I imagine most US ones wouldn't qualify...).
As for why they bother... it's not like a patent. If you don't defend a trademark, you lose it. Granted this is kind of different in that the original trademark was set out by international law, not a trademark registry, but the principle is the same. If they don't go after people, it sets a legal precedent, so they'll have a harder time fighting someone who's abusing the symbol.
I don't believe they've sued anyone to date, because every company who's gotten a letter (that I know of) has adjusted their symbols to comply...