Will hosting a blog to your own server help optimize it better for search engines vs using their free server.
Doesn't matter really. It's just better to have it on your own server cause you actually own that web hosting, rather then the hosting company closing down their "free" service and you losing your site.
In other words, if your question is "Should I use a .blogspot.com blog?" then NO!!! As Voasi said......you're at the mercy of Blogger shutting you down for being a splog. Even if you aren't. I'm a control freak, and would prefer to manage my own stuff.
Man - sound like words out of my mouth. Yeah - why not spend the $6 a month for hosting and get your own domain for you blog?
Get your own domain and Word Press. It's a bit better for the SEs, it won't go down on you, it's easier to remember, and looks more professional.
If you use blog on your server it will help in your pagerank and rankings of your main site. If you host it on a free server, your own site does not get any advantage.
Less than $10 a month for hosting, which generally includes a free domain name. 1000s of hosts will do this for you.
You could always do both. Post a bit of your blog at blogger and link to your site for the rest of it. It might bring you more traffic that way.