Renaissance Faces: Van Eyck to Titian, National Gallery, LondonAnnie Leibovitz, National Portrait Gallery, London

The National and National Portrait Galleries have always made odd neighbours, much of the latter's collection belonging on the walls of the former. Every now and then, though, their dual existence makes sense, as with this week's (entirely unintentional) pairing of Renaissance Faces at the National and Annie Leibovitz at the NPG.

Renaissance Faces is pretty much what you'd expect from its title, being a study of European portraiture from 1425 to 1575. The period saw the greatest explosion in Western thought since ancient Athens, a revolution fed by technology and trade. Hand in hand with a change in man's view of himself went new ways of expressing that view. A classically based humanism moving north from Italy collided with a vogue for oil-based paints moving south from the Netherlands. The whole act of making art – the idea of what art was for – was revolutionised in a generation, and nowhere was this clearer than in portraiture.

All this is familiar enough and, indeed, there is little in Renaissance Faces that will surprise – certainly not the works in the show, which come predominantly from the National's own collection. The subject is broken down into sub categories – Families, Love and Beauty, etc – and the idea that artists worked with the simple aim of reproducing what they saw in front of them is challenged. In this slightly pedestrian portrait of portraiture, the aim is as often to disguise as to reveal: thus the Metropolitan Museum's copy of Jan Gossaert's Virgin and Child is a depiction of Adolf of Burgundy's wife and son, while a heap of vegetables by Arcimboldo turns out to be Emperor Rudolph II. Pontormo's Portrait of Two Friends is the depiction of a thing – friendship – rather than of a person, just as Palma Vecchio's Portrait of a Poet is as much a picture of beauty as it is, allegedly, of Ariosto.

All of which is to say that portraiture is a word that, with a little gentle pushing, can be made to creak at the seams. In the NPG's Leibovitz show, it can even include photographs of gardens with nobody in them, although it is the identity of the nobody in question that matters. The garden is the one at Charleston, and the person who is so achingly not in it is Leibovitz's dear, dead friend, Susan Sontag.

Leibovitz's portrait of Sontag's absence is among the most eloquent in this moving show, in part because, unusually, we cannot question the relationship between the photographer and her sitter. Another example is in a small image of Sontag, cut up and pasted together over what looks like a second chopped-up photograph. What goes on between Leibovitz and her subject can only be guessed at, and that is because Sontag – eyes closed and in her best clothes – is dead.

If there is a problem with Leibovitz's public photographs, it is that she is now at least as famous as most of her sitters. The notorious case of Regina v Leibovitz – the Buckingham Palace portraits are by the door to this show – suggests that the American photographer is not unaware of her own royal status. She shoots Daniel Day-Lewis or Mick Jagger or Colin Powell as equals, their celebrity being reflected in her own.

Heisenberg noted that recording a thing necessarily changes it, and Leibovitz's ease with her subjects does the same. Her shot of Mikhail Baryshnikov on a beach is unsatisfying when compared with an adjoining one of Leibovitz's old mum, also on a beach. The latter is funny and painful and tender while the former is merely elegant, the icon of an icon by an icon. Pictures of Patti Smith and her sons or the Clintons back stage lack edge because they lack awe. The last section of Renaissance Faces deals with portraits of rulers, and Leibovitz does a nice line in these herself. Unlike Titian or Antonis Mor, though, she is also a queen regnant.

What gives this show its power is her personal shots, never before shown in public. These reveal Leibovitz's genius to lie in something that looks very like love – for Sontag, for her own children (Leibovitz took time to photograph her babies as they were being born), for her parents, her heroes. When she shot the veteran photographer Richard Avedon – then nearing 80 and frail – he fretted about the result. Later, calmed, he wrote to Leibovitz: "Thank you for taking care of me." Care, perhaps, is what makes her great.

National Gallery/National Portrait Gallery, London WC2 (020-7747 5930/ 0844 579 1924) to 18 Jan/1 Feb 2009