Several interesting and true replies to the original question have been posted so far. I have flown Boeing, McDonnel Douglas, and Airbus products, in addition to an assortment of turboprops.
It's not a question of which is the better aircraft. I currently fly an A330, but would stick with the B777 if I had my pick. Boeing's might use some old technology, but that's not always a disadvantage. According to my experience Airbus builds very smart airplanes for very smart pilots; while Boeing builds quite smart airplanes for quite dumb pilots. I consider myself an above average stick & brain, though far from flawless. I have found myself mentally in row 45, though I was physically strapped into my pilot seat. In those cases it has been an advantage to be able to easier (i.e. less demanding on the old noodle) transfer through manual control to full automation with a complete awareness of the situation and the systems.
Therefore, I think the Boeing line of products are superior to Airbus, ONLY because it's more of a pilot's airplane than a system operator's airplane. I am absolutely not talking about actually handflying the airplane. I'm talking about autopilot control, systems integration, and flight control computer programming issues. A side stick is perfectly fine, and ALL big airplanes are "fly by wire" anyway. All you do in any big airplane is hydraulics actuation anyway. Also, even Boeing would have used a side stick if United's design team hadn't intervened. After all, a stick is the most natural way you can think of to control an airplane.
However, the question is most likely in regards to what type rating a jet newbie should look for. Here's what I think:
Avoid paying for a type rating if you can. There are many companies who employ guys with only turboprop experience. If you are trying to jump ahead of the game, you might be fooling yourself. Companies like Southwest Airlines (in the U.S.) employ only pilots with 1000 hours P.I.C. Turbine. That means that for some jobs it might be better to stay with a regional airline and make Captain, and then transfer to jets. It'll also help when you are transferring to the left seat in a jet.
If you still want to get a type rating, here's the skinny. Get the rating that will make you desireable for as many companies as possible. Sit down and look in JP Airline Fleet to see who flies what type of airplane. Big operators will not care what rating you have since they're going to type you anyway. Look for small, upstart type carriers. They're the ones most likely to give you a chance if you have a rating in your pocket. If most of them fly Boeing, get a B rating; but if most of them fly Airbus, get an A rating. That simple.
Good luck.
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"Good judgement comes from experience. Experience comes from bad judgement."