I have never heard that theory before today. It does make a bit of sense so I decided to Google it. When I typed in "p tag vs" the auto-complete came up with "p tag vs br tag" and I added in "seo" to the search. There were only two related results.
1. http://www.seo-works.com/p-tags-in-seo.php While the article presents an interesting idea, they offer absolutely no references for the discussion at all. The author's name is not listed, no date, etc. This is where I begin to look at the site itself and it seems like a "build your website in 30 minutes" type of creation where I keep the idea (good ideas can come from anywhere) but I lend absolutely no weight to this article as evidence.
2. http://www.pitstopmedia.com/sem/br-hr-p-pre-tag-seo This page isn't that great either, but it is a huge improvement over the other result. We can see an author name "TraiaN" but it is not a link and there is no information about the author. It has a publish date of Nov 2010 which is good.
What bothers me is the information is presented as factual. It is presented in the same was as if I was to say "the sun is hot". Sure the sun is hot. Everyone knows the sun is hot. But I am not willing to accept the theory that p tags are superior to br tags on the same level.
The second site does reference a search patent from 2004 as evidence to support the statement. The particular section of the patent that is referred to only talks about the desire to apply weighting to words near header tags (h1-h6). Never once is the p tag nor br tag mentioned.
These type of articles really muddy the SEO world by presenting one person's theory as a fact. Neither author has presented even one shred of evidence to support this theory. Neither claimed to have worked at Google and seen the algorithm or have performed any testing. There is no evidence whatsoever that either author has any knowledge about SEO. This is the SEO equivalent of snake-oil sales.
Summary: I am not aware of any evidence linking any weighting to text contained within a
tag versus text separated with a
tag. I suggest you use the best programming practices within your site. Follow the W3 industry standards when coding your site and use tags appropriately.