Are there any advantages to freshly ground salt?

Not really in terms of flavor, no. Salt doesn't have a substantial number of volatile compounds that are released by grinding, in contrast to something like whole peppercorn, which releases a range of aromatics when the outer hull is breached. Even specially-sourced salts which contain a range of flavorful minerals don't really release them into the air, they dissolve into the foods they're added to instead, and/or are picked up by your tongue as you eat.

One advantage grinders offer is that a lot of them are adjustable, and can be tweaked for everything from a very fine grind to large crystals. This can be helpful depending on the effect that you're going for - seasoning popcorn with very fine salt vs. garnishing a steak with large crunchy chunks, for example. Smaller crystals with more surface area will generally tend to be perceived as saltier due to more taste bud contact, and will also dissolve into liquids (i.e. soups or stews) more quickly. I like using larger crystals sometimes for flavor/texture contrast and visual appeal for some dishes.

Another advantage mentioned by others is that finely pre-ground salt may also contain anti-caking agents to keep it from clumping together. If that bothers you, a grinder gets around this because larger pieces won't stick together as much before being pulverized by the grinding mechanism.

Some producers package salt in disposable grinders just as they do with pepper, and they share the same problems: cheap grinders simply aren't very good at getting a consistent grind size, or they have mechanisms which wear down quickly. Reusable grinders are often much better, and as you've seen they're getting very easy to find even in non-specialty stores. I like ceramic grinding mechanisms personally, because I feel that they get a better "grip" on the particles and are better at consistency, but good steel mechanisms will do well too. Here are some suggestions for care to help you get the most out of them.

I personally like having a salt grinder for popcorn. When I salt my popcorn, I want the crystals to be totally pulverized with a much more fine grind than is available in the box.

Finely ground salt tends to cake without additives, so putting rock salt in a grinder will give me pure salt dust with no additives.

Another good reason for having salt grinders is that they are FUN!

Perceived quality definitely affects our enjoyment of food and drink, plenty of research has shown this.

But another factor in meal enjoyment is how fun it is to eat. Many ways we eat involve doing things that are not strictly necessary for flavor, but still make the process more enjoyable. Kids walking around gnawing on giant turkey legs understand this. It's more fun than eating from a plate. It's not just the food, but the experience of eating it.

At the dining table, there are many simple procedures the diner can use to transform the food in some minor way at the table.

Dipping chips in a bowl of cheese, loading up your fondue fork and dropping it in the pot, sprinkling some cheese and red pepper on your pizza, composing your own salad at a salad bar, even things like adding sugar to tea or coffee, all provide an opportunity for the diner to influence the final product.

That guy who always puts salt on everything? It's only partly about the saltiness, it's partly about the ritual.

Salt kept in a coarse form can be stored without needing any anti-caking or similar additives (check the ingredients list on a pack of table salt, there are often other mineral additives for that purpose), since caked salt is a non-problem with a grinder (unless it is wet).

And as mentioned above, being able to control the fineness of the grind is advantageous, and allows you to only store one style of salt and having it available ground as needed.

The main point for me is missing from the other answers:

Very finely ground salt has a larger surface area (due to the smaller particle size and irregular surface) than "normal" table salt. This makes it dissolve more quickly.

When completely dissolving the salt (e.g. in a soup), there will be no difference between finely ground salt and coarse salt. But if you use a grinder to add a bit of salt to your dish at the table, you will need less salt for the same taste. And also, depending on the grinder, be able to distribute the salt more evenly.

Verifying what I just said is pretty easy: Taste a grain of coarse salt and the same amount of finely powdered salt. The coarse salt will taste less salty but for a longer time and the finely ground salt will have a really strong taste but be gone more quickly.

If your home is humid, the salt will start to clog. This makes it more difficult to serve properly.

Grinders solve this issue.

Style points! I would also guess you might buy higher quality salt to go in the grinder.

That aside, if it's a good grinder, it will allow you to select the grind size, which you might want to do to add a bit more/less crunch to your salt.

There should be no difference in freshness or taste.

Salt is salt, and as you said the granule size isn't important if you're cooking with it (usually). It matters when you add salt at the table, though, where a finer salt is normally preferred.

Especially, the size does make a difference to how the salt soaks up humidity from the air. The finer the grind, the faster the salt will become "wet" and start to stick. Coarse salt can last for months and years with no issue, but fine salt can start to clump after a few weeks in a humid climate (like where I live).

To solve this issue, the manufacturers put an anti-caking agent into the finely ground salt, which the coarse salt doesn't need.

Some people (myself included) prefer not to eat the anti-caking stuff. To solve the problem of humidity, we buy coarse salt and use a grinder to get fine salt on demand at the table.

It's not about being freshly ground, it's about making high-quality salt which comes in large crystals useable at the table.

My thoughts on the actual differences have already been detailed in other answers. But I actually wished I had one handy a few days ago, so I wouldn’t have to buy a box of “table salt” (fine powder) which I hardly ever use. What’s left in the formal dinnerware’s shaker is all I have. Just grinding the kosher salt flakes, if easy enough, means not needing that.

Oh, there might be a difference: I recall seeing a show on candy where different rock salts were used as they have different flavors. Having your choice of that, in a powder, would be available with a grinder.