In terms of SEO, the last I heard there's no difference between:
blogger is different though.
However, I think you got the main purpose of the blog wrong, you should focus on the value the blog might bring to the business and more importantly to its customers, not whether it helps in ranking the site better in Google.
If the blog won't be of any value to its customers, then don't put it on the main domain. A blog can be a powerful branding tool, but this diminishes significantly if it's not on your own domain and even more if the design doesn't match (or at least reflects closely) the business website and overall branding.
In short, any SEO activity should be part of an overall, holistic and integrated marketing strategy, and not a stand-alone tactic.
There's much more to it, but I don't have the time to go into it right now.