Again, you are conflating Congress with individual members of Congress. Congress has absolute immunity from the Executive and the Judiciary because, as it turns out, they have the legal weapons to dispose of the persons representing the two other branches.
An individual member of Congress shares in that immunity until expelled by either House. You'll notice that in France they extend an even grander sort of immunity to the President. They won't be able to prosecute Chirac until later this month. Our President, on the other hand, has an unlimited power to pardon ~ presumably Bill Clinton signed one to use "just in case" just in case the he got "convicted" in the Senate in his impeachment trial. He'd need that done before he was kicked out!
Congress as an institution has that immunity. Individual members of Congress do not.