I've never heard the expression 'give off the impression'. As Yannis said - give off means that something emanates or radiates from something. So - it's OK to say 'give off a bad smell' or 'give off heat'. To me, that doesn't work with 'impression'. It may be a phrase that is used in other countries?

The difference lies in the words "give" and "give off". "give" means exactly what it means: to have something and pass it over to someone or something else. "Give off" is used when something radiates from a person or thing. For example, the sun gives off light. Now, both sentences essentially are saying the same thing. I would say that using "give off" is more dramatic that would mostly be used when describing objects rather than people.