The Typhoon and Rafale are like teenage brothers. They like to be competitive but hate to admit that they're really quite alike.
If there's a big difference, the Typhoon is more of an air-to-air aircraft and the Rafale is more inclined to move mud. Surprise, surprise: when they got started, which of the sponsors had a relatively new bomber in service?
This relates to another difference. The 'phoon is really, really good at supersonic turn and burn, better than the
PR machine gets across. This led to some very laborious development in flight control, but it seems that it is sorted now.
On the other hand, the Rafale can really carry freight - it's a bit like a 25 ton supersonic A-4 - so if you want to go a long, long way with big gas tanks and a load of weapons, it's your airplane. The Typhoon's better than you might expect, but can't quite do what the Rafale does. And the videos of the Rafale barreling through the Pyrenees at 5.5 g and 600 mph with a load of iron are quite impressive.
Whatever the French word is for bibbedy-bobbedy-boo, the Rafale's stealth relies on it. The Typhoon is front-sector RCS reduction plus jamming, a reliable technique if not as flash and mysterious.
Both aircraft can land on a carrier but the Rafale can do it more than once...
The French tried to get cute with the radar and backed a dead-end idea, especially combined with the restricted nose volume that is characteristic of most non-huge carrier jets. They need an AESA.
The Typhoon's radar reminds me of the $6000 turntables that you find in audiophile stores because some people don't trust CDs yet. But if it works as well as they say, it does some things better than an AESA, and does a lot better than the sort of AESA you could have baselined in 1996-98.
I bow to Officer Viking's comment about the cockpit, with the observation that it's French. That is, it's a bit like getting into a Citroen DS in 1957 and going "why the *&^% did they do it like this?" and then realizing it makes sense. The oddest feature is what looks like a What The Butler Saw machine sticking out of the panel.
The choice between the two comes down to price and (even more importantly) support, which is why 'phoon salesmen remind you every two minutes that they have contracts for 600-plus aircraft.