What are the benefits of dried ingredients?

If I'm not mistaken, you're probably cooking the Cantonese dish "braised chicken with dried shiitake mushrooms/'winter' mushrooms". Yes, using dried shiitake mushrooms (in Cantonese, also called "winter" mushrooms) is quite essential to this recipe. The sun-drying process breaks down proteins to amino acids, so it increases the umami flavor of the mushrooms, providing strong flavors that could not be matched by fresh shiitake mushrooms. For some references, see this page.

In Chinese cooking, dried raw materials are often more prized than their fresh variant due to the higher intensity of flavor. For example, dried scallops vs fresh scallops, and dried abalone vs fresh abalone. The dried variant provides much more intense flavor, and is thus much more expensive.

If you really can't find dried shiitake mushrooms, you could use way more fresh shiitakes, maybe squeeze the water out of it to reduce the volume. You could also use other means to make up for the lost amino acid content by using chicken stock, or, er, MSG. However, the amino acids in chicken stock and/or MSG wouldn't be the same as those in dried shiitakes, so the flavor would be still somehow different. Yet another alternative is to enrich the flavor in other ways: for example, add some Cantonese barbequed pork belly meat to the braise to create a thicker flavor (as is done for many Cantonese braises).

Not sure about any particular 'benefits' of using dried. I only ever used to use dried when I couldn't get fresh. These days I can get fresh shiitake in any supermarket [London, UK].

Shiitake are a fairly 'meaty' mushroom anyway. From dried you need to soak them in boiling water to scare some life back into them [& I'd definitely keep that soaking water with the potential to use is as extra 'stock' if needed]. Fresh have a 'bite' to them, for sure, but dried definitely have a 'chew' too. I used to end up throwing half the lower stalk away as being too tough to want to eat.

I honestly can't remember if there's a significant flavour difference, but if you've ever had mushroom ketchup, that's how I recall the taste of dried. It's somehow more 'fermented'. Maybe in some dishes, that's a desired flavour profile. It's certainly not 'bad' in itself, merely different.