As a fan of anime and manga, my time and money are precious resources. Since they are limited, the question becomes “Is it worth experiencing the same story in another format?” The answer is complicated with plenty of exceptions and personal taste.
The first thing to clarify is that I am going to focus on adaptations where the story is practically identical. There are many adaptations where the creators decide to go a different direction to the manga. In these cases, you are experiencing a new story with different plot threads and arcs. Famous examples include the 2003 adaptation of Fullmetal Alchemist where the latter half of the show is different from the manga and the 2001 series of Hellsing which featured stories not found in the manga.