For me marriage is very important, it felt like the natural next step to show each other our commitment, to me it solidified our family (we were engaged when we fell pregnant and DS was 3 months when we married). I never see marriage as just a piece of paper, the vows mean something to me and it means a lot that my husband has said legally and spiritually that he intends to be with me forever, after getting married the only way I can describe it is security, I felt secure with my position in the world, in my relationship...it made me feel very...content I guess you could say, it's made us stronger because any argument we have had and to be fair we don't argue much it never gets blown out of proportion, we deal with the matter at hand because we know we will work it out becuase this is forever, none of that dramatic stuff you get as kids (we've been together since we were 17 so it's the only relationship we've had so I guess that does skew my beliefs as I've not had a grown up relationship that wasn't a marriage lol).
On a more superficial level Im a traditionalist and I guess I just like that we all have the same name, that my children have happily married parents (emphasis on happily, no point being married if you're not happy!!) I'm proud of that.
Practically it's good to be married too, financially, and as we are a military family it's very important, the military is old fashioned still and basically completely overlooks girlfriends no matter how long you've been together or got kids, to be a Next of kin (not just a point of contact) during deployment you have to be married (or family) to be seen as a dependent by the military to get housing and support you need to be married, so on that level it's important for that too.
And on a really shallow level, I just love ticking the "Mrs" box lol.
Strangely enough neither our sets of parents or my grandparents had good marriages, both ended in divorce and my grandparents were bitterly unhappy, but if anything we have learned from their mistakes, we know why they were unhappy, it wasn't because of marriage it was just they picked the wrong people, they didn't spend much time for each other so I've learned a lot from them and am keen not to make the same mistakes, but if anything coming from families with weak relationships I think it has made us all the more keen to provide a strong happy family unit with a happily married mum and dad as we know the benefits. DH's grandparents had the idyllic marriage, I think that's what encouraged my DH, for me it's my best friend's parents, they have a fantastic marriage and she's the most mentally sorted person I know lol, I can really see the differences in us and how our upbringings affected that (just want to add I had a fantastic childhood I don't mean that I didn't!)
In terms of if one wants it and the other doesn't I've not had to deal with that, I don't think marriage is of quite the significance to my husband as it is me he doesn't over think things like me lol, but he always wanted to get married so it's all good. If someone didn't want to marry I don't think I could be with them, purely because if marriage wasn't important to them we would have very different values so probably wouldn't be a suitable couple anyway and would clash in other areas particularly child rearing I imagine (only because of how important marriage is to me).
But this is what works for us, I put my hands up and say that I am usually an insecure person and I think marriage was a form of security for me, but I don't judge if anyone else does things differently in this respect, it's much too personal a decision and it's 2015 where thankfully we have choices unlike 100 years ago, but I think marriage makes *my* family stronger.
Sorry that was quite a long reply........!!!! It's a subject I'm quite passionate about I guess.
To be fair, not much in our relationship has changed since getting married. Although there's a certain bit of certainty in our lives because we made that commitment to each other. 
