[quote=Kestrel]In the future; don't let players argue amongst each other.[/quote]

Easier said than done. A DM holds the power and responsibility over if something is accepted or not, but they're not the roleplayers parents. A DM can speak or advise against conflict all they want, but if players are in the mood then it will happen, and nothing the DM can do will stop it. You're the story teller and/or organizer, but not the care taker. If your group of players are constantly bickering it's probably a sign your players just aren't a good mix.

[quote=Rilla]You handle it simply.Tell them both that you'll be making decisions on the character, and that your word is law. If the first one leaves, after you accept the 'similar-but-not-quite', that's probably someone you don't want anyway, as they could easily turn into the type of player that threatens to leave any time they don't get their way.If you think the second one is too similar, request the changes to make them different and see where it goes. Essentially, you're the GM, your word is law and they have to respect that. Just stand up to them, as otherwise, they'll run all over you... repeatedly.[/quote]

This, combined with Captain Jordan's post below...

[quote=Captain Jordan]Finding a balance between this, and being a dictator, is the crux of any good leadership. Good luck![/quote]

Is basically what I would be advising. 

As the GM you have a certain amount of responsibility on you to keeps things working, but you can't always be held responsible for players either. Sometimes you just get a bad egg, [u]however[/u] and GMs get this confused with "The GM is never wrong/always right", get rid of this mentality, now. I've dealt with many GM's and Admins who acted as if they had all the power and were not to be questioning. One was an D&D campaign who drove his campaign to the ground cause of it (and out of spite sabotaged the following one out of revenge), other's simply felt they could dictate every last thing a person was allowed to say or do (Both IC and OOC), or ran a site as an admin/mod with a similar mentality. Those last two cases without fail would see the majority of their players gather up and leave elsewhere as a result (often on account of me saying "Fuck this, let's go somewhere where people are treated fairly" sometimes on account of others). 

Now, you may be going "That's all nice, but how [u]do[/u] I act then rather than how don't I act?

For that I'd simply pay a few moments of attention of the admins of the Guild. They're some of the only people in charge of such a huge community where even after constant evaluation I have yet to find anything wrong with, and I'm someone who on more than one occasion has a horrid GM/Leader and as a result of their bad leadership got everyone under them to follow me elsewhere, where people wouldn't be ass holes (and they'd all be thankful for it in the end). 

Basically, establish that you're the one in charge, that in the end you're making the final shot. But listen to your players, find the issue, see if you can find a solution that works for everyone. If you can't do whatever seems the most fair/just. You will still make enemies, you will still get the odd player storming off, but you should have the majority of people under you happy with your leadership. People willing to go the extra mile to help you because they're happy with you in charge, rather than simply sitting there obediently until someone can convince them to move elsewhere.