Text figures flow better in running text. When the numbers are part of a prose sentence and when you aren't comparing them with other numbers or doing math with them, they can be more readable (and less jarring).
Since we don't have old-style numbers in web typography today, it's understandable that they look weird to you, but they aren't a case of different- for- difference- sake.
The option to use Old Style Figures should always be applauded. As is obvious from the case of Georgia, it’s never good when you are stuck with only one or the other.
I think they're fantastic for addresses and phone numbers. Distinctiveness is a win there, since the numbers are abstract and would otherwise just run together.
The '8' in for instance "Suite 1850" is a little bouncy.
As far as it being easier to read in running text, I can't see it, but if the designer of a given work truly feels it provides that benefit, then so be it. I just would prefer for this style to not be used on the web, even though it is becoming available along with other advances in web typography. I just think it looks bad and it's hard to read.
But having said that: the kernel of easy- to- agree- with in your comment is, for a lot of the figures that tend to get typeset on web pages, lining figures are the more appropriate choice.
That said, when set in running text and when describing things that are, effectively, proper nouns (IP addresses, street addresses, phone numbers), old-style figures often read better and are more attractive.
You should read up on typography before decrying things about it. Robert Bringhurst’s The Elements of Typographic Style or Warren Chappell’s A Short History of the Printed Word are two great places to start.