As an example, If there was a horn antenna with a large aperture operating at microwave frequencies, and in the aperture there was a UHF dipole antenna at the opposite polarization, with a separate feed, that would be a shared aperture antenna. An antenna with one feed that is able to operate at two frequencies is a dual band antenna. This could be a slotted microstrip patch for instance.
Perhaps the only real difference is whether the diplexer is in the aperture or somewhere back in the feed network, because usually the two different frequencies have to be separated somewhere.
I agree with Malcolm White , but in my opinion, the term "Shared aperture antenna" is more general.
In the general case, Shared aperture antennas are the recent trend of antenna technology that integrate the usefulness of several antennas for radar, communication, navigation, electronic warfare, and Synthetic aperture radar etc. applications into one aperture using multiple band and multiple beamforming technologies.
In this regard, the term "dual band antenna" is the part of the modern term "Shared aperture antenna".