2. I am on two weeks' holiday.
When "two-week" is used as an adjective, as it is in Sentence 1, it takes a hyphen and remains in the singular. There are many examples of this:
It's a twenty-metre drop.
He's a seven-foot giant.
So - "I am on a holiday which lasts two weeks" becomes "I am on a two-week holiday".
"I am on holiday for two weeks" becomes "I am on two weeks' holiday" (no article required).
The first sentence specifically suggests an actual holiday, a trip abroad or at least a journey away from home. The second sentence could simply mean someone is not working for two weeks. We would still refer to that as two weeks' holiday, even if you sit at home and watch TV for two weeks!