The officer advising Tharkin to evacuate wouldn't be the first guy in such a position to be ignored by the man in charge, for better or for worse. AFAIK, we don't know that guy's credentials and rank. Tarkin is likely convinced that his Death Star is invulnerable to fighters (in 99.99% of the situations that likely is the case too). Now he gets a guy telling him it's not and he should evacuate. Either this guy has found, on the fly, a weakness that the entire design team missed or he's just overreacting under combat stress. Which one would Tarkin consider more likely? Moreover, if he evacuate and everything proceeds as planned, he has just lost a ton of credibility.
Additionally, not launching fighters at all might also be an attempt to sideline Vader and keep the entire glory of ending the Rebellion to himself.
Piett at Hoth: there's no indication he was aware the rebels had an ion cannon. There's also no indication his fighters were in the hangar as opposed to dispersed to cover more potential escape routes from Echo Base. In the absence of the ion cannon, Piett's ISD can more than adequately handle one transport and 2 X-wings.
Edited by LordBlades