Both words are synonymous, but there might be a bit more intelligence implied by the word intuition.
"Instinct" literally applies to non-learned behaviours. Animals are governed by instinct far more than humans are, born with sometimes highly complex behaviours such as migration patterns. Most things considered human "instincts" are just survival reflexes. Instincts are also usually fixed patterns of behaviour that do not allow for free thought. Of course, the word is used in a broader, figurative way to describe human behaviours or choices which seemed correct perhaps because of transferrable knowledge or skills, which does imply some level of learned skill or intelligence.
"Intuition" involves applying logic and intelligence to a new situation. For example, computer operating systems are sometimes described as "intuitive", meaning that someone who has never seen it before should find it easy to use because it is designed in a logical or perhap familiar way. Although the word does mean the ability to do something without relying on too much thought or skill, it implies a little more intelligence than "instinct", and is perhaps more synonymous with what some people call "common sense".
For your example, I would use "instinct". Waking up to an alarm or some other sound meant to wake us, when we perhaps sleep through other ambient sounds, is very much an instinctive response.
As an aside, you may also want to look up pavlovian responses, which are a "learned" response to a particular stimulus (eg an alarm sound), although I don't think it would fit your example without considerable adjustment.