Apple,
this is a great question which will keep doing the rounds forever. I personally don't think there is an answer to it.
All airlines have totally different selection procedures. And all airlines more importantly have different Chief Pilot's and DFO's. Therefore all airlines in my opinoin have different tastes about what they want.
An integrated course will prove to an airline (if you do very well) that you have handled a large amount of information and testing very well, and also in very short time. You will have also completed the same course that many airline Cadets go through.
A Modular route shows different valuable disciplines, such as being able to work and study AND learn new motor skills all at the same time. These are indeed very flexible and valuable assets.
However, I went down the integrated route and was lucky enough to gain a place on a competitive Airline placement programme after my integrated course ended. I got the feeling that both the training provider and my airline liked the fact I had gone to Oxford and had done reasonably OK.
It gives them an audit trail if you like. Lots of reports and people that they know to check me against.....all very reassuring.
I felt (maybe incorrectly), that a Modular route was scrutinised very closely in every area. This is because a modular route is much more difficult to check up against....due to the fact that training elements may take place in different establishments.
All I can say is that I was integrated and I really enjoyed it. It was good for me. I am flying a 757 as my first job. This I guess doesn't happen that often, but it DOES happen from time to time. So, which ever route you choose, just try your hardest, be nice, make contacts and don't shut any doors on yourself.
I paid my (lot) of money, and luckily I got what I dreamt of at the beginning.
At the end of the day, I am still convinced that most of comes down to having a positive mental attitude, being average (not being a stunt flyer), right place right time and a whole load of luck at every stage!!!
GOOD LUCK.