Does amblyopia actually hurt or is it just a discomfort?

You can't feel it. Amblyopia cannot be felt. It is poor vision that cannot be corrected. There are a number of causes, the most common being crossed eyes or asymetrical refractive errors.
No. Amblyopia is often not even detected by younger patients. There is no pain involved. The blurred eye may even be ignored by the visual system and not even be noticed as very blurry although the vision is very poor. Sometimes though, both eyes are very poorly seeing and eyestrain may be present since neither eye is clear.