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I had never thought of the similarity 
You're close enough, though.
Languages change and erode over time (such is life), and words acquire curious new meanings depending on the context they're used. Cretin used to mean "Christian", and silly was an old word for "blessed". And the same happened here with bizarre consequences.
The Latin word hospes could be used to refer both to the guests (reason why the Portuguese word for guest is "hóspede", from the accusative "hospitem"), and to the host (the word eroded a bit more in English, as you can see from the bold bits in the Latin example I posted).
Now, Latin had a suffix fetish, adding suffix after suffix as it saw fit. If you wanted to turn "hospes" into an adjective (i.e. the state of having guests, the quality of having guests), all you had to do was drop the final "s" and add "-(t)alis" so it could become an adjective. If the words "guestly" and/or "hostly" existed (my auto-correct thingy says they don't), it would mean the same "hospitalis" meant. A guestly place was called "hospitale"... and, unless you're a doctor, a nurse, or Jenny, you're pretty much a guest in a hospital. You come, people take care of you, and you sod off.
Remember I said Latin was crazy about suffixes? For whatever reason, they felt the need to turn the adjective into another noun, despite the fact it had already come from a noun. If it sounds stupid to you, just keep the word "manliness" in mind, and you can see how "manliness" and "man" don't mean quite the same thing, the former being more abstract than the latter. As I said, they had "hospitalis" and felt like turning it into a noun. "-itas", modern English "-ity" was the right suffix for the job. All you had to do was drop the final "s" and, voila, the property of having the quality of having guests (now that's a mouthful!) became hospitalitas - or, as we know it in English, hospitality!
Because words get a life of their own once they receive new affixes, "hospital" became the building where "guests" entered all the time; and hospitality became something else altogether. But, yeah, if your hosts don't take care of you, you might need a doctor to do just that 
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