The arrival of Rosencrantz and Guildenstern Are Dead at the Old Vic at the same moment as Travesties is running at the Apollo means that suddenly there are two plays by Tom Stoppard running in the West End – the first his calling card of brilliance, the second the effervescence of his middle years.

What a pleasure it is to have them back – and particularly to see Rosencrantz and Guildenstern returning to the theatre where its first National Theatre production made such an impact 50 years ago. Stoppard has such a distinctive voice as a playwright – clever, witty, endless playing with concepts and conceits, effortlessly profound.

The old view of him was that he was all head and no heart. But what David Leveaux's production of Rosencrantz manages to do, is to keep all its sharp energy and smartness, yet simultaneously remind us too that this is a play about two men who are about to die. A sense of sadness and loss underlies the jokes almost from the start and so a play about two bit players in Hamlet is as stalked and marked by mortality as Shakespeare's tragedy itself.

Yet it is astonishingly funny with a glorious range of jokes from slapstick (the pair's failure to catch Hamlet with intertwined belts) to conceptual – Rosencrantz cries fire in a crowded theatre because he is "demonstrating the misuse of free speech" – to the magnificently ironic such as The Player's exasperated exclamation: "We're actors, we're the opposite of people." The dexterity of thought dazzles.

The production is also beautifully cast. In Daniel Radcliffe as Rosencrantz and Joshua McGuire's Guildenstern it has two protagonists who are like a matching salt and pepper set or non-identical twins, alike, interdependent but ultimately differentiated by subtle markings. As they swap syllogisms or play endless party games, their need for each other to make sense of a world in which they are not even sure they exist binds them close. McGuire covers his rising panic with a nervous smile and a constant reassuring pat of his hair. But it is Radcliffe's Rosencrantz that is the revelation – endlessly trusting, with wide, gentle eyes full of bafflement yet occasional moments of clarity about the depth of their plight. It is a lovely, confident performance, a million miles from Harry Potter.

They are balanced by David Haig's charismatic The Player, a man at once swaggering, rueful and vaguely threatening. His pent up power lights up the stage and his delivery of lines such as "You should have seen us in better times. We were purists then" has a marvellous dying fall, as he addresses his raggle, taggle troupe, all stylised white faces, tattered costumes and mournful sax. The nature of acting, and the role of theatre, is just one of the many rich themes of the play and it is embodied here by the way the players' style is different from that of the actors who are playing Hamlet and different again from Rosencrantz and Guildenstern themselves.

The whole thing is framed in a slightly over-emphatic design by Anna Fleischle. I liked the way it uses the entire depth of the Old Vic stage, bringing the action as far forward as it can, while placing its effects at the back, and dividing the space with silk curtains but it occasionally over-complicates the action, particularly in the second half. Leveaux's direction, on the other hand, is carefully modulated, pulling the different threads together with considerable skill.

Rosencrantz and Guildenstern Are Dead runs at The Old Vic runs until 29 April.