Let there be absolutely, positively no mistake here. "Tags" have completely different uses depending on the platform on which they're used. We're going to take a look at three of the most popular platforms and how tags affect them: Wordpress, Ning, and Tumblr.
Before we dive into each, let's get one thing out of the way. Those who say that tags are old and no longer useful are simply being lazy and encouraging the same. It takes less than 30 seconds to come up with a handful of appropriate tags to go along with any blog post and therefore it falls under the category of "why not?" Nobody outside of Mountain View, CA, knows for certain how tags benefit search engine optimization. They do, however, definitely have an opportunity to benefit the reader. It's a best practice that is getting pushed aside by many. Don't fall into the laziness trap. Tag away!
Tags for Wordpress
Depending on how you have your site indexing set up in the back end, tags allow search engines to find similar articles. The two major types of taxonomy, tags and categories, are intended to help people navigate a blog. As a result, Google and Bing will follow tags and categories in order to see what level of understanding a blog has on each individual topic.
The tag pages themselves, once visited, will give the search engines a depth-of-content picture. For example, if you have a blog for a Nissan dealership that often uses the tag "Altima", the search engines will be able to see that you have written a good amount of content on the topic. Many would argue that they know this already and that semantic indexing is designed in part to replace tagging as a method of establishing authority, but again, "why not?" It definitely doesn't hurt to tag. It probably helps on Wordpress, even if only a little. There's not reason to skip the few seconds it takes to add them.
Tags for Ning
On the Ning social platform, tags work in ways similar to Wordpress, but with an added bonus. Blog posts on Ning do not have categories the way that discussions do. As a result, tags become the primary taxonomy that search engines and readers use to navigate a site when they want to see similar articles.
Some have also speculated that there is a direct SEO value to the individual post, that the search engines look at tags very similar to how they look at highlighted content and will give a post a lift in the rankings as a result. This is unconfirmed and I've never tested it myself, but I would speculate that it is true.
Tags for Tumblr
This is an entirely different ballgame. On Tumblr, tags are everything. The community lives off of tags in a way that is similar to hashtags on Twitter, but there are certain ones that are eternally "trending", so to speak.
Tags are Tumblr users' primary method of discovery. Tumblr has devalued them a bit in recent months and focused on "Spotlight" that highlights individually influential tumblogs rather than the community-rich "Explore" page that lets people surf tags, but they're still extremely important and can help a new tumblog get found by the community. Getting found on Tumblr is the key to both social exposure and search; reblogs by other Tumblr users are extremely important and can mean the difference between having an invisible tumblog and having an extremely popular one.
Tags for Humans
At the end of the day, the real benefit of tags should be for the readers. Google and Bing may or may not pay attention to them, but allowing your blog visitors to latch onto a particular topic and follow it all the way through is a way to make your blog stickier.
Again, it takes second. Why not?