Consider this first person sentence:
I stepped out my door, felt the wind brush across my skin, and knew it would be a beautiful day.
Then consider this third person sentence:
He stepped out his door, felt the wind brush across his skin, and knew it would be a beautiful day.
There really is very little difference, at least in this single example. But the connection here is not simply in the way the words are arranged. It is that in both examples, the action is perceived through the point of view of a single character. Changing the I to a he didn't really alter the story. He still stepped outside, felt the wind, and drew a conclusion about how the day would unfold. The same conclusion. Both examples also deny the reader access to certain things, most noticeably the inner thoughts of other possible characters.
So first person and third person have a lot in common. But there is a key difference, and this difference involves the reader's distance from the point of view (POV) character. In first person, you're right in the mind of the POV character. You get the joy (or the agony, depending on the book) of having access to all the POV character's innermost thoughts. Because of the nature of first person narration, you get all the thoughts of that one character. All of them. It can be fun. Or very annoying, depending on how the author handles it all.
Third person, on the other hand, gives the author a few options. It can be very much like first person. If you replace the pronounces (and alter your grammar here and there), first person becomes third person. You can still access the innermost thoughts of the POV character and get to enjoy the same privileges you had in first person. But this is only one way to handle third person.
You can take a totally different approach to third person and lock yourself entirely out of the inner thoughts of the POV character. You may have no idea what he's thinking or why he's doing the things he does. You're simply following the action. This can be fun and a bit exciting, but it can also be irritating. Sometimes you simply have to know what the POV character is thinking because he's acting like a little idiot and all you see are the actions. The thoughts are hidden. In the hands of a master, this point of view works well. With a novice...not so much, but that can be said with most points of view.
Between these two extremes, there are a dozen other options for third person point of view, and most fictional stories fall somewhere in the middle. Close enough to satisfy but not close enough to irritate, as one of my creative writing professors once said. But remember that as you get further from the character, third person had less in common with third person.
So which is best? Neither, really. It depends on your story. But it should be noted that one of the greatest benefits of first person or close third person is allowing you to understand a character who isn't all that verbal. So if your main character doesn't like to talk? Break out the first person (or the close third person, if you like). Your narrative will be all the more special for it because--let's face it--everyone thinks.