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I do not have perfect pitch. I have been playing piano, playing sax, and studying music for almost 20 years. I have familiarized myself with music enough to have developed relative pitch. I can find my way around any song when listening to it, I just don't know off-hand which key it is in.
All this to say, I have a potential student wanting to learn piano for the purpose of learning perfect pitch. I cannot decide how to reply.
I think it is cool some people have this naturally, but I can probably count on one hand the times I would have needed perfect pitch.
What are the benefits?

I would recommend reading the excellent Wikipedia article on Absolute Pitch. There is no significant indication that it can be taught.

My own experience: absolute (perfect) pitch has definite drawbacks. The "A" known as such to one possessing this ability is linked to a particular number of vibrations per second (Herz). If one's "perfect" A equals 440 Herz, then exposure to variations (higher or lower) will cause distress sufficient to negate any previous advantage. Such variations can be historical (Baroque tunings), regional (local traditions of tunings higher or lower than 440), and situational (a cappella choral performances where the pitch drops during a piece).

Of far greater value is the development of relative pitch that you mention. Aural recognition of scales and intervals is essential. Ditto knowledge of chord progressions. And, if you are a singer, comparing an unknown pitch to your own vocal range can be a quite useful tool, as the vocal muscles do have memory.

Perfect Pitch is a human ability very similar to Relative Pitch. Both are usefull in playing-enjoying music and both have to do with the way someone hears. But, perfect pitch is mistaken sometimes as born-wth ability or othertimes as less usefull.

More or less, I will gently disagree with jossuk. There are no drawbacks in learning Perfect Pitch. There are benefits, that together with relative can develop a very good hearing ability!

These are : 1. to be able to hear what's the different between pitches (why A sound like an A, what Bb sounds like, why that music piece is writen in that tonality, what happens in music when you tune higher or lower) , 2. to be able to identify pitches without a relative one (that means you can tell what a pitch is without starting with a known one, because of no1), 3. to be able to sing pitches without get a known pitch, 4. improves your musical memory (because your mind has another way to remember music). That what I can think for now.

And for the drawbacks, let me explain. If you have both relative and perfect pitch, there is no music confusion. There is no "I can't stand listening to this music because it is tuned lower", because the only thing you hear is pitches, you can not only tell what's reference between them, but also tell what the sound like. In other words, you feel like tuning yourself lower and listen that way.

Of course, it can be tought! To be precise: of course, anyone can recall that ability, because it is something we are born with, but none tell us to practise.

I think I helped!

Aris, your gentle disagreement is noted. Perhaps I wasn't precise enough in my posting.
(1) Please read the entire Wikipedia article on "absolute pitch"; otherwise, we're not close to sharing the same basis for discussion.
(2) The article seems to confirm my own experience, as I have not met any musicians who claim to have developed "perfect" pitch. Any testimony from musicians claiming to have done so would certainly be of interest.
(3) I have worked with musicians whose "perfect" pitch (innate) was not at all helpful in the situations I mentioned earlier (another point of agreement with the article) and I respect their conclusions. Again, it would be interesting to hear from anyone with contrary experience, i.e., that absolute pitch was ...always... a benefit, no matter what the local circumstances.

I found this old discussion by accident, but actually I have something to add to the discussion, namely that absolute pitch to some extent can be developed. As a teenager I was tested by a music teacher from the local 'seminarium' (which in Denmark is a school where you educate teachers, not a religious institution), and I hadn't got anything like absolute pitch at that point.

Then I learned to play the violin and the cello, and I composed music like a madman for a number of years, sitting in an armchair with a fountain pen and some hectographed paper because I didn't have a piano yet. Later on, when I built my music collection, I decided that I wanted a collection of themes for all my vinyl records and cassettes, and if I couldn't get the scores from the library I had to write the themes down myself.

And after many years doing this I can now see the notes when I listen to music, and if I then find the scores afterwards (for instance at IMSLP) I almost always get my suspicions concerning the tonality confirmed. Part of this is based on many years acquintance with instruments (I have some idea about fingering patterns on many different instruments), but I have checked my accuracy on a site that only presented the testees with isolated tones played on a piano, and there I was never more than a half note off - which actually is something that according to one expert in the field may happen to elderly people for physiological reasons. But when there is real music involved I can compensate for this because I know the instruments.

So when is absolute pitch a liability? Well, some organs are tuned too low, a few orchestras are slightly too high, and both things irritates me. But it irritates me even more when I want to write down a theme from a piece that is supposed to be written in E flat major and I hear a piece that to my ears sounds like D major. Then I sometimes try to tweak my own perception, but with complicated music I may have first to write the themes down as I hear them and then (reluctantly) transpose them to the official key.