The Movie: It begins with an elderly Bilbo Baggins, the same one we remember from the Lord of the Rings trilogy, being played by Ian Holm. An Unexpected Journey uses this bookend as a narrative device, having Bilbo writing the "true tale of events" in the Red Book of Westmarch, including (eventually), the famous "In a hole in the ground there lived a Hobbit" opening line of the book. He addresses Frodo as he writes, making it seem like the book is being written for his nephew/cousin/whatever.
I had about the same reaction at seeing Frodo.
Frodo and Bilbo have a short exchange about an upcoming party (Bilbo's 111th birthday), and the scene ends with Frodo scampering off to wait for Gandalf, precisely where the beginning of Fellowship of the Ring picks up. After Frodo leaves, Bilbo sits down for a smoke, and then the story starts proper - with Martin Freeman playing a younger Bilbo.
The Book: Very little is actually changed; instead, Peter Jackson has just added an extra scene. The only change worth noting is the great age difference between Bilbo now (played by Ian Holm) and Bilbo then (played by Martin Freeman). He looks significantly younger as Martin Freeman.
What does it matter? The aging of Bilbo Baggins, or lack thereof, is actually an important plot point (both in the books and in the movies). The One Ring grants extended life; one of the things that tips Gandalf off about Bilbo's ring is the fact that he hasn't aged. The addition of Frodo and the "post-adventure" Bilbo has no impact on the written story.
My Opinion: Regarding Bilbo's aging, I'm not too concerned or upset by the change. Hobbits usually don't make it to their 111th birthday, and showing Bilbo looking as young as he did (comparatively speaking!) for such an advanced age is no big deal. He did age, and noticeably so, but certainly not six decades worth. An error this size is a small price to pay to see Martin Freeman's excellent portrayal of Bilbo Baggins. However, I'd honestly rather have just started with a younger Bilbo Baggins in the first place; there was no need to show Frodo and Bilbo talking at all. Tying the movie directly into the opening of Fellowship was not only unnecessary, but it dragged the movie on before it even had a chance to pick up.