What is the difference of Feshbach resonance and Fano resonance?
In physics, resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonant frequencies, or resonance frequencies. (Copied from Wikipedia: Resonance.)
The Fano resonance and the Feshbach resonance are the same thing, when it comes to their mathematical essence. Both of them describe the resonances with the asymmetric shapes and were derived during some research of the coupling between the bound states and continuum. Sometimes the term "Fano-Feshbach resonance" is being used.
Herman Feshbach (1917-2000) and Ugo Fano (1912-2001) developed their descriptions independently in the late 1950s and early 1960s. Feshbach was thinking on nuclear physics; Fano on atomic physics. More general comments on these resonances etc., see this paper.