Privileges are, well, privileges you earn depending on the amount of reputation you have. As you earn more reputation, you will be able to do more with the site. At the early stages it's mainly about basic participation, but at higher levels the privileges are more moderation-centric. A complete list can be found in the Help Center: Privileges.
Badges are just small rewards for doing (hopefully) good things. A complete list of these also appears in the Help Center: Badges.
More specifically, tag badges are rewards for contributing good answers to questions in specific tags. In each tag having at least 100 questions it is possible to earn three different tag badges:
- Bronze. 100 total score for at least 20 non-community-wiki answers to questions in that tag.
- Silver. 400 total score for at least 80 non-community-wiki answers to questions in that tag.
- Gold. 1000 total score for at least 200 non-community-wiki answers to questions in that tag.
(Note that this is score and not reputation earned; each upvote contributes 1 to your score, not 10.)
Which one is "more important" is a completely personal matter. Some users have no need/desire for privileges beyond the basic participation level (which max out at about 2000 reputation), and for them tag-badges (which somewhat demonstrate mastery in a topic) are perhaps more important. Others are more interested in taking part in site moderation, and for them the higher privilege levels are more important.
(It should be noted that with a gold tag-badge comes a sort-of privilege: the ability to immediately close a question as a duplicate as long as it was given that tag by someone else; or to immediately re-open a question with that tag which was closed as a duplicate.)