Question as in the title.
I commonly use the phrase "out and about in town" in speech.
I'm not sure if my usage is correct because of the "night on the town" phrase.
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Both usages are correct.
The usual phrase for being in the city is "in town". This just refers to your physical presence there, without much implication about what you might be doing there - one would normally assume that you were shopping, visiting people, or even just staying at home (if you happen to live there).
The phrase "on the town" has a slightly different meaning - there's no one specific connotation, but one would understand that you were visiting restaurants/pubs/bars, socialising, and very likely drinking as well. I doubt that there is any direct etymological link, but to me the use of "on" is similar to usages such as "on the prowl", "on the warpath", "on the booze", "on the pull" and so on.
"Night on the tiles" is a phrase which draws imagery of the wild nocturnal activities of cats as they have fun over the rooftops. As metaphorical expression are used and adapted over time they adopt new meaning and connotation, so what was a "Night on the tiles" has evolved to become a "Night on the Town".
Simplistically, the meaning and intention of the phrase "Night on the town" is easier to understand than the phrase "Night on the tiles". It is implicit in the former that you will be going to town at night, in the later you may have to explain what exactly you mean by tiles.