I don't know if it's right place to ask it, but I searched google and books the answer I get seems to be like resonant antenna have small bandwidth and have standing wave instead non resonant antennas have wider bandwidth and travelling waves. What I don't get that how should it affect my choice of choosing one over other for a particular situation, like what I lose when I choose one over other and what I gain.
A non-electrically-resonant antenna is when the antenna is used at its optimum "natural" frequency. Example: a half-wave dipole where the incoming signal wavelength corresponds to twice the antenna length. In these circumstances, the dipole appears as a resistance fairly close to 74 ohms and, either side of its "natural" resonant frequency, it might look a little capacitive or a little inductive but, the main component of its impedance is resistance. This makes it difficult to directly electrically tune to gain any benefit and, because of this, it has a naturally "decent" bandwidth.
Another example is the quarter-wave monopole antenna used on FM radios to receive 88 to 108 MHz broadcast stations. These are non-electrically-resonant and have a decent bandwidth.
A good example of an electrically-resonant antenna is the old-fashioned crystal radio antenna. Typically it's a monopole but, because it's being used to receive transmissions in the 1 MHz or lower part of the spectrum, it's physical length is well-short of a quarter wavelength and, because of this, its dominant impedance is capacitive. The received signal voltage is also significantly smaller than a proper quarter-wave monopole but, because crystal radios use a tuned inductor, the arrangement of short-antenna (dominated by capacitive reactance) and the tuning inductor magnify the low signal voltage significantly. Together they are an electrically-resonant circuit. They can be "selective" enough to tune across lower the frequency bands and receive each transmission without much co-channel interference.
The decision to choose one over the other is usually down to accommodating the space required for the antenna. If there isn't much room to have a full-length "natural" antenna, then you will choose a "short" antenna and possibly implement a direct tuning regime to select the desired station.