The fact that they are separately stored in browser cache will cause mysterious results, when the blog is viewed alternately using the two different aliases.
This is not a terrible problem for your readers, when they simply read the blog. It can cause problems with the search engines, though, as they index your blog based upon inlinks from your readers.
When you publicise your BlogSpot blog, you are hopefully careful to use only one of the aliases, consistently.
If this blog was published only to "bloggerstatusforreal.blogspot.com", that is the URL that I would use. But there are reasons why Blogger provides us with the use of the "www" alias too, and that has very likely spawned a community of bloggers who refer to their blogs using the "www" alias.
If your blog is inlinked by a blogger of the latter community, I will bet that some of them are similarly inlinking to the "www" alias on your blog. So now, you are going to have some readers inlinking to the "root" of your blog, and others inlinking to the "www" alias.
What do the search engines do? They index based upon the inlinks, then they see duplicated content between the two diferent addresses, and they levy the infamous "duplicated content" penalty. Worse yet - because of the asynchronous cache issue, they're going to also index different content occasionally.
Google has provided us with a solution to this "split personality" problem. If you setup Google Webmaster Tools for your blog, click on the "gear" icon, and select "Site Settings". For "Preferred domain", you'll have the choice to have your blog indexed consistently, as either the "root" or the "www" alias.
If you specify your preferred domain as http://www.example.com and we find a link to your site that is formatted as http://example.com, we'll treat that link as if it was http://www.example.com.
You should note that, if you're using a custom domain published blog, as in "www.example.com", this option is more likely to work if you setup the domain DNS properly.
A very simple setting, easy to make, and with apparently magical results. No more schizophrenic indexing. Simple, no?