The useful model: Antenna bandwidth is the frequency span meeting a chosen criterion such as SWR or gain; efficiency is the fraction of accepted power radiated. They are related through antenna Q and loss but are not interchangeable.
What controls the result
Adding loss often broadens the SWR curve while reducing radiation. Electrically small antennas can be efficient only with very low-loss conductors and matching, which commonly leaves them narrowband.
The misleading shortcut
A broad 2:1 SWR specification can look superior even when a resistive or lossy network is responsible for the bandwidth.
Make the better decision
Define the bandwidth criterion, measure or estimate efficiency and gain across it, inspect component loss and heating, and compare antennas at equal accepted power and installation.
Evidence boundary
Propagation and antenna metrics are meaningful only with their frequency, path, geometry, reference plane, loss assumptions, and measurement method. Use current observations and the exact installed system.
