I believe you to be referencing two related yet different things: A metadata tag and a social hashtag.
A tag is a non-hierarchical keyword or term assigned to a piece of information (such as an Internet bookmark, digital image, or computer file). This kind of metadata helps describe an item and allows it to be found again by browsing or searching. Tags are generally chosen informally and personally by the item's creator or by its viewer, depending on the system.
Hashtags are mostly used as unmoderated ad hoc discussion forums; any combination of characters led by a hash symbol is a hashtag, and any hashtag, if promoted by enough individuals, can "trend" and attract more individual users to discussion using the hashtag.
While the social hashtag may have been adopted as "geeky way" to trend relevant social posts together, I would not say that the tag has evolved into the social hashtag.
Update: I kind of jumped the gun on the submit
Now that that's out of the way, let's answer your question. I would say how you handle "tags" depends on which you are handling and how they are used. For instance:
Let's say we are a social network where users post a common message such as:
"Fan of @coolband? Come grab #beers with us #downtown tonight! #comeonecomeall #allthefunz"
In this scenario our "tags" are integrated into the message, so we won't want to pull them out completely. We could pull out the "#" and, assuming they were live-links, unlink them and put them at the bottom. But we also have tags hanging off the end which, arguably, were meant to be displayed to enhance the core message content.
Our example app here uses these tags to add enhancements/playfullness moreso than to help categorize our message.
On the contrary, let's say we have a full blog post app. Here, we write our post about our vacation where our friend nearly fell into the Grand Canyon. It's text content, but our platform is not used in a way where we would add "hashtags" into our long-form post. However, we can and do add tags to the post. Perhaps they are descriptive like "vacation," "grandcanyon," but lets say we also add "lol" and "myfriendsaredumb."
While you could argue that the latter two were added to add playfullness to the post, it is still our app's intention to use these tags as categorization. It is here, that we would have a more keyword/search related tag experience and I would expect the UX and design to dictate that.
These are really two different ways we use "tags;" each based on how our app mostly interprets the use and purpose of each.