Alfa,
In a nutshell, integrated courses train you up from zero hours to CPL/IR fATPL level at one training establishment (or partnerships) and it takes a little over a year, start to finish, full time. However it is generally the more expensive option, but it is said that airlines look more favourably on it (hence Apple's question).
Modular is as the name implies. You can complete different modules (ie. ATPL theory module, CPL flight test module, IR module) at different training establishments, in your own time (obviously complying with the CAA/JAR completion deadlines) and it allows people to hold down a full time job at the same time (social life ? what's that then ?). You will need a PPL to start the modular route (hence quite handy for us guys already with some experience) and generally 150 hours to undertake the CPL stage.
As is evident, there are pros and cons to both systems, the likes of which as already stated, will be debated ad-infinitum, but it is generally hoped that both will be recognised by airlines "reasonably" equitably in the future. Just a question of time I hope. We'll see.
Hope that helps.