These two idiomatic expressions have significantly different implications...
I know where I stand
I have a clearly-defined opinion (on some contextually-relevant topic).
Often carries the strong implication that I don't want to discuss the issue - because I've already made up my mind, so there's no point in talking to people who only want to persuade me to change my mind.
Effectively My mind is made up, and you will not convince me to change my opinion.
I know my place
I realise that I have low "social status" in the current context.
Often with the implication that I shouldn't say what I think (or even contribute to any debate on the current topic). I am expected to simply endorse the opinions of those with higher status and/or more power than me.
Effectively I am servile, and must do/think only what I'm told to by my superiors.
Given the preceding context (I shouldn't have talked back to him), the first example would normally be understood to mean that I shouldn't have wasted my time talking to him. Obviously he only wanted to engage me in debate in hopes of persuading me to change my position, but I was never going to allow that to happen anyway, because my position is "fixed, not up for debate".
The second example would normally be understood to mean that it was socially / politically unwise of me to disagree with him - because he has more power than me, it was dangerous / impertinent of me to challenge him.
EDIT: This question obviously generates a lot of interest, so I thought it might be useful to consider the far less common assertion I know my position, which could be used with either of the above senses...
Personally, I feel that the two different idiomatic meanings aren't entirely arbitrary - the reason they've been assigned this way is at least partly because to stand is normally a voluntary act (often against resistance - consider withstand, stand up to, take a stand). Whereas to be placed [somewhere] is effectively a "passive (servile)" action.
As a class-obsessed Brit, I can't resist adding a link to the classic I know my place comedy sketch (from The Frost Report, 1966, with John Cleese, Ronnie Barker, and Ronnie Corbett), where physical height is amusingly conflated with social status. British humour at its best!