And this is why I hate the anime so damn much. That is the only time Dante was a hobo or anything of the sort of what you are describing was then and then after that damn thing, suddenly, that is the only way people see him. And a burn-out? Since when?
Seriously? This is how you saw him in DMC1? A slouge, good for nothing, lazy burn-out? Our perceptions of Dante in that game couldn't be any more different than if you told me he wasn't wearing red, it was polka dots.
Well, not so much killing people, not anymore. By the time Nero gets there everyone's already dead and I'm assuming Dante got there after him.
I've been studying a bit of history and you'd be surprised at how important looks are. It's why kings and presidents look the part while their country might be in poverty, because if you ruler, the man that represents your nation and your people, looks like a hobo it would speak volumes about your nation.
In the same way Dante always strides to be an imposing figure. In the novel it mentions how he tries to keep up appearances, to create and effect. So, yes, his looks are important to him.