They are not one and the same.
Immigration officers have much wider ranging powers. A normal tam ruat will not check your 'status' - I doubt that most would know how.
ID is sufficient for police (or Immigration if you can also show something regarding your status), as said, if they want to see your passport you have 24 hours to present it at a police station.
Whilst I have read stories of aliens being taken to a police station while a friend retrieves their passport, I have never heard someone claim that it actually happened to them.
For 17 years I have taken my passport out with me. I have photo ID and carry a passport copy in my wallet. I only take my passport if I am going out of province - and then it stays in a hotel safe. Never a problem.
I don't expect a problem in the next 17 years either.