This isn’t a passive game. You should actively compare the brand stories they are telling to your own. Put yourself in the place of a potential customer – who would you be more likely to do business with when you visit either company’s web site?
How do they describe themselves? Are they using the same language as you? When you visit their web site, is it clear what they are offering customers? Now take a look at your own site. Is it clear what you are offering customers?
Sometimes, it’s helpful to take a look in the mirror to see who you are and how you might be perceived. Sometimes, though, it’s helpful to sneak a peek over the neighbor’s fence.