'So am I' means 'I am xxx, too'. For example: 'I'm thirsty'. 'So am I. Let's go and get a drink.' 'So I am' is a less usual phrase, but we sometimes use it mean 'You are right! I am xxx.' 'You're covered in pencil shavings!' 'Oh, so I am! I didn't notice. The lid must have come of my pencil sharpener. How annoying. I'll just go to the wastepaper bin and shake them all off. Thanks for telling me!' Be careful how you use these expressions, because they aren't interchangeable.

So I am would be used more for confirmation of something that's already been asked/said. Example: " You're an American, right?" Answer: "so I am", meaning "that's right". On the other hand, so am I means "I am as well". Example: " You're in the military? So am I."

"So I am" is an Irish filler phrase. Such as "I am very tired, so I am" is just unnecessary emphasis on the subject I. Or a regional way of saying "yes I am" to a question like "Are you tired?" Or a statement "you look very tired" I am native English and would never use the phrase. I would avoid it. If I used it, people would wonder why I was trying to sound like an Irish person.

Sorry,I meant "What's the difference between 'so I am' and 'so am I' ?"...