David,

My guess is that a person with perfect pitch would identify 440 as "A" and 443 as an "A" out of tune.....too high but not necessarily 443.

They would never think it was a "B" at 494.

The difference between 440 and 443 is 11.8 musical cents.  Even I can hear 11.8 cents but, in my case, I would not be sure what the note is. 

A good (old time) piano tuner is reported to be accurate to 10 cents and more recent studies say 5 cents.  I'm not sure of the full abilities of someone like that.

In all my years I have only known one person who had perfect, perfect pitch.  A young high school girl who played piano.  She also may have been the best interpretive musician I ever heard and I'm not kiddin'

I can't identify individual notes but I am pretty good at guessing keys. 
For instance, the key of "A" sounds "happy" to me.  Maybe it's the way that my violin plays ?   "Ab" is "serious" but pleasant, etc.  Maybe most beautiful.  "D" is good fiddlin' sound, etc.

Incidentally, I am finding that I am more and more able to look at a piece of music and hear the tune.  I can identify a lot of music without a title.  I think with a violin I often resort to relative pitch ( 3 rds, 5 ths, etc. )

I know that is small potatoes as talents go BUT what amazes me is that I could not do that years ago.  I learned that somehow.

I guess this what untapped abilities are all about.  Not apparent until used?