Right - If the frequency response is similar they will sound similar. There are other differences (polar response, sensitivity, etc.) but the basic "sound character" is frequency response.
You would think that everybody wants a "flat" microphone, but it turns-out most people don't like "perfectly flat" and (slightly) different curves sound better with different sound sources.
Of course, you can adjust the frequency response with EQ so I think people go a little too crazy trying to find the best (or better) mic. If you're working in a pro studio with lots of mics available, of course it's nice to choose the best mic for every situation.
I'm not advocating using a "cheap mic" but you usually don't need an exotic mic either.
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You should easily hear more bass in the large diaphragm
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That's a good generalization with large & small diaphragm mics but it depends on the particular mics.