First of all, you are asking two questions in one. The first being the difference between can't say/can't tell in the first example, the second being the difference between being unable due to lack of knowledge vs prohibition (the second example).
In the first case (lack of knowledge), the difference is:
He may or may not be injured, but I can't tell.
You have not been able to glean any information yourself. You just don't know.
I think he might be injured, but I can't say for sure.
You may have some idea, but you cannot say with confidence.
In the second case (prohibition):
I can't say, I promised.
The normal usage, means what you expect - I am prohibited from saying/revealing that information.
I can't tell, I promised.
I would say this is incorrect in the case of prohibition - I have never heard it used in this way. This would be probably misinterpreted to mean some other valid use of "tell", such as:
I can't tell her, I promised.
In this case, tell means to "say to someone else".
I can't tell on her, I promised.
This has a completely different meaning based on a completed different definition of tell - to "tell on" - to reveal the wrongdoing of someone else to an authority. I.e. a full sentence might be:
Who stole your Apple?
It was Elise.
You should report her to the teacher!
I can't tell on her, I promised.