Maybe I'm starting some 'wild academic d8scussions' :)

The simple answer is to be found here:  https://en.wikipedia.org/wiki/Current-feedback_operational_amplifier

Basically it is the negative input that makes the difference between these two types: high impedance for voltage feedback, low impedance for current feedback.

Current mode (or current-feedback) opamps are opamp circuits in which the main amplifying block is a transimpedance amplifier rather than a voltage amplifier as in voltage-mode opamps. The non-inverting input voltage is connected to the input of a voltage buffer stage (high input impedance, low output impedance, unitary voltage gain, so i+=0). The output of the voltage buffer is connected to the inverting input so that ideally v-=v+,  and i- is the actual input (error signal) which is fed to a transimpedance amplifier. The output voltage is fed back to the inverting input through the feedback net and if the loop gain is sufficiently high, the inverting input current is almost zero as in voltage-feedback opamps.

Current feedback opamps can be employed almost in the same negative feedback configurations of their voltage feedback counterpart, the most important difference being the fact that closed-loop gain and bandwidth are not traded off in the same manner.

Generally speaking, current-mode opamps in negative feedback configurations show a worse accuracy (higher offset voltage, lower DC gain, worse noise figure) and a better high frequency behavior (large bandwidth, high slew rate) compared with their voltage-mode counterparts.

You can find more details in many online tutorials, e.g.:

http://www.analog.com/media/en/training-seminars/tutorials/MT-034.pdf

Very basic considerations, see appendix below

One thing that need to be watched out is stability. The feedback resistor used in CFOA-based amplifier cannot be zero (In case you wanna make a unity-gain buffer). It needs to be higher than a certain value. Read the data sheet carefully. 

...and for the same reason, there must be no capacitor in the feedback loop (unless it has a series resistance with a minimum specified value).