Your professional artist bio is kind of like your résumé in paragraph form (but less boring). It highlights your top accomplishments—usually with the most recent and most important at top.
Your bio gets to the point. It's not a place for you to share everything you've ever done or get into your personal life.
Your bio is written in 3rd person unless it's an autobiography, in which case you'd use the 1st person.
Your artist statement is about your art, not about you. More to the point, it's about the current direction of your work, not a history of how you got to where you are now.
Your artist statement is written in the 1st person. After all, it is a statement.
For good measure, let me throw in one more piece of writing.
About pages are the page on your website where your artist bio and statement come together. They should be less formal than the proper bio you would use to share with a museum or gallery.
How you approach your About page depends on your goals as an artist, but there should be text about both you and your art on this page. I'm fond of About pages that use humor combined with an interesting twist on serious facts. I tried to do that on the Art Biz Success About page.