I am using OS 10.15
Is there an advantage in going to 10.15.1?
Yes, there are always reasons to go to the “point updates” – generally bug fixes, but also security updates, which were part of 10.15.2 which was released yesterday:
Goodness, yes. We’re still recommending that people do not upgrade to Catalina yet, but if you’ve taken the leap, absolutely install all the bug fix updates as they come out.
Although I tend to agree with the advice to go ahead with the point updates, I would still hold off for a few days. These past few months we have seen several point updates where things got screwed up big time for a select few only for Apple to then release a point update to the point update or re-release an update to fix whatever they had broken last update. The people who suffer through those shenanigans are the early adopters. I prefer not to be one of those. I usually give it a couple days. I figure my risk of having an exploit used against me during those 2-3 days is much smaller than that of me getting burned by a bad update the likes of which we’ve unfortunately seen several from Apple during the recent times.
Of course the irony of that thinking is not lost on me. If everybody shared that attitude nobody would run into issues and we wouldn’t be warned about them, let alone Apple made aware that they goofed up an update. For that to happen somebody has to take the plunge and suffer for the rest to benefit. I’ll go ahead and admit that in this case I’m selfish enough to hold off and let others take that hit. 
I’m confident there are presently still more than enough people who have set up their Macs and iPhones to auto update.
The head of the department in my first DP job had the attitude of wanting to upgrade just behind the ‘bleeding edge’.
Yep, that’s always good advice. We install all this stuff as soon as it comes (and we have time to reboot), and there will always be enough others on the bleeding edge to take the cuts. 
I did not upgrade to 10.15.1 but have since upgraded to 10.15.2
It is faster than 10.15 and I am pleased with its performance.