Yes! Actually, I use the two (about myself) somewhat differently. I use "older" when I'm speaking more relatively, recognizing that many people like you are older than me but that I'm up there on the age spectrum.

I use the word old when I wish to sound more blunt, especially when I'm comparing myself to my younger self.

I mean, I'm certainly old now compared to when I was in my 20s or 30s. So, in context, old can work better to convey not merely a chronological meaning, but an emotional meaning.

But it's a little different when I'm writing about someone else. I would be very reluctant to write something like, "Michael Horvich is old."

Perhaps in the right context, it would make sense to write that sentence. But it seems unnecessarily reductive and less than fully respectful.

But when you write it, it has a different flavor and feeling, without smacking of condescension or disrespect. After all, you live in your own body, and if you want to acknowledge the fact that it's old, well you should certainly know, and I should respect your observations.

That's my take anyway.