Right, let me try and sort some things out. New Order, Depeche Mode and The Cure are all great but New Order are better than the others because they have been more adventurous musically (have Depeche Mode or The Cure ever produced anything like Blue Monday, Thieves Like Us or Fine Time?) and have made more great albums (Low Life, Brotherhood and Technique being the three most obvious choices). However, what Depeche Mode and The Cure do better than New Order is produce great singles that appeal to people who aren't loyal fans of the band (such as DM's "Just Can't Get Enough", "See You" and "People Are People" or The Cure's "Love Cats", "In Between Days" and "Friday I'm In Love"). Depeche Mode's and The Cure's singles compilations are excellent but you wouldn't really recommend buying all their albums (I haven't got any other Cure albums but I've got all DM's albums and I would only recommend buying "Violator" and "Songs Of Faith And Devotion" and the live albums). New Order release singles (or, more accurately, their record company releases singles and no one else is convinced that they know what they're doing) but they don't produce many singles that appeal to a wider audience and as a result are not as famous as DM or The Cure.
Leaving The Cure aside, I'll try to sort out New Order vs. Depeche Mode. New Order are a group of relative equals and they are at their best when they are all on the same wavelength and/or pulling in the same direction (which hasn't happened often since 1990) but that means that when they are at their best you can't really pick one member and give them all the credit. Whilst Depeche Mode were a four-piece and are now a trio two members of the band have always been more important than the others and they are the songwriter (first Vince Clarke and then after he left Martin Gore) and Dave Gahan (who has gradually developed from being the lead singer of a synth-pop band to being the frontman of a stadium rock band with synths). If those two are on form and working together well DM can make brilliant music but if one of them is off form or they aren't working together well they can't. However, what Martin Gore can do that no other member of DM and no member of New Order can do is take musical and lyrical ideas and hone them to perfection. Martin Gore can write gorgeous stripped-down ballads and huge storming anthems combining synths and guitars (I wonder who first thought of combining those two?) that ideally suit Dave Gahan, DM and huge stadiums full of screaming leather-clad Americans or Europeans who want to hear classic Depeche Mode. Like Noel Gallagher Martin Gore doesn't do many different things but when he's on form he does the few things he does do brilliantly.
To sum up, all three are great bands but of the three only New Order are a genuine group (DM are Martin Gore's band which Dave Gahan fronts and The Cure are Robert Smith's backing band) and can do what all the best bands in the world do which is become more than the sum of their parts.