The main differences between the two versions are as follows:
1. The earlier version was shot in black and white.
2. The cast was British/European, starring Edna Best, Leslie Banks, and Peter Lorre.
3. Instead of a young son, the couple (Bob and Jill Lawrence, not Dr. Ben and Jo McKenna) had a teenage daughter, Betty.
4. Jill, an expert markswoman, shoots a villain (I can't remember if it was the always incredible Peter Lorre or a henchman) off of a rooftop in order to save her daughter.
5. The story starts in Switzerland, instead of Morocco, before moving to London.
Technically, the second is regarded as the more complete film, what with its better constructed sequences of suspence, the hit song Que Sera Sera, and Jimmy Stewart and Doris Day in the lead roles, which have been further complicated and developed from the original. However, I personally like the story of the earlier version better, since the wife is more proactive in the couple's efforts and because I'm not horribly fond of Doris Day as an actress. My film professor, who has a somewhat unhealthy obsession with her, thought that her persona as America's Sweetheart was completely subverted by this film. That still didn't make it any more watchable for me! Also, oddly enough, it felt like the second film took longer to play out than the first - and I'd seen the second earlier than the original.
I recommend watching both, if you can, and deciding which you like better for yourself.
Also of note: This was Alfred Hitchcock's only official sequel, though a lot of his films are patterned after The 39 Steps, including The Lady Vanishes and North by Northwest.