if i get a new hosting and a domain name, is it still the same blog?

[Mark]

as far as the search engines are concerned, is it the same blog or a new one?
what happens to the old blog, do i have to close it, or does it change into a new one?

[Sirn]

New hosting will only affect your blog if the new host has settings that alter you content (some content might not work or will show differently). A new domain name will for sure be a different blog unless you import all the files from your previous blog or redirect your old domain to the new one. Closing the old blog is up to you. As for search engines, it'll take time for search engines to register your new domain.

[Jake]

As far as search engine ranking it would be pretty much restarting from scratch as a new site, Google can't distinguish a relocated site from an unauthorized copy made by someone else.

A big help in maintaining continuity is to maintain control of the old domain name and redirect it to the new one, this might involve setting up an .htaccess file that has a redirect for each individual page, a 301 permanent redirect is honored by the search engines, backlinks would effectively be redirected to the new site as well.