This is by far the superior version!
It's far clearer on the reasons why Harry resigned his commission, for a start. The characters have proper conversations about it, and they include the scenes from the book of young Harry listening with horror to the old soldiers talking about battles, and then going to look at the portraits of his military ancestors which are all over the house. Harry in the 2002 version, to me, wasn't convincing in his reasons at all.
Also, Harry's father in the 1939 film dies before he resigns his commission - I had wondered how Harry had been able to afford to get to Egypt in the 2002 film, since he'd lost his officer's pay and presumably his father had cut him off without a penny. In the 1939 film, he'd inherited his father's property, so had funds available.
The 1939 film also gives a good reason for Harry managing to disguise himself as an Arab without learning Arabic - he's supposed to be from a tribe who have had their tongues cut out! Harry has a clearer plan of action too, though the character of Abou Fatma is absent, so he has no help to carry it out.
Ralph Richardson is very good as the officer blinded by sunstroke, and the 1939 film actually shows Harry getting him from the desert back to the British fort on the Nile, rather than jumping straight back to England.
The battle scenes are epic in scale. They were even filmed at the original locations of the battles, in the Sudan. Apparently there were extras there who actually remembered the battles. The 2002 battle scenes, especially involving the British soldiers forming a square, were also very good, though the film has been criticised for dressing the British soldiers in scarlet jackets, when they were actually wearing khaki at the time. The 1939 film gets the uniforms right.
The 1939 film was set ten years after the 2002 film - the Royal Cumbrians were sent to the aid of General Gordon in 1885, while the Royal Surreys were sent to be part of Kitchener's army in 1895, which meant that the escape from the prison at Omdurman was timed to coincide with the battle of Omdurman.
It was strange to see John Laurie playing the Khalifa, though. I don't think a modern film would have cast a Scottish actor in the role.
It's also quite strange to think that British forces, and their allies, were fighting the Italians over the same terrain just a year or so after the 1939 film was made.