Both of them help you to organise content. Both of them help you get the right kind of traffic on your blog. So yes, they are important.
Let me take the example of my blog and help you understand the difference between the two. I have categories called Poetry, Life Musings, Short stories, Moments (photography-based), Quotes, Short Stories, Snippets, etc. on my blog.
Use your categories for broader topics.
I can check all the poems on my blog at randomnessinked.wordpress.com/category/poetry
I have more specific tags called nature poems, self love, life. People who are interested in reading about self love will find your post when they search for the tag in the WP Reader.
Use your tags for diving deep into the theme of your posts. Be specific. Be emotional.
I can check all my posts tagged as life at randomnessinked.wordpress.com/tags/life
You can have subcategories too. Scroll up to the menu of the Community and hover your mouse over Events. Monday Meet, Tell Tuesday, Throwback Thursday and Saturday Share are sub-categories of the category called Events.
You cannot have sub-tags.
WordPress suggests bloggers to use a combination of 5 to 15 tags and categories. If you cross 15, your posts will be marked as spam in the WP Reader.
Got to know that Askimet suggests 5-9 tags and a maximum of 11 tags.
Check out some popular tags on WP here.
Use your tags and categories wisely!