Supercapacitor or battery with capacitor
Start by reading about how to use the capacitor charge, current and energy formulae. Q=CV, Ic=CdV/dt, E= ½CV²

Then realize that Batteries have more capacitance per volume than  super caps, but only above the final Vf cell voltage, so stored energy is E= ½C(Vi²-Vf²).  

Current is limited by the ESR of the charging circuit and Cap/battery. Parts with lower ESR are generally rated for higher currents but this value changes with voltage and aging.

example photo not to same scale
-- >3kJ supercap vs  >50kJ

This Supercap with a cost of $100 for 1000F with 10mΩ ESR has a T=RC=1second with a much faster response time than a smaller LiPo battery which has greater energy storage capacity.

This LiPo has much more Capacitance value but also much higher ESR.

E= ½C(Vi²-Vf²) for C=1000F and Vi=2.5V and Vf=0.0 (complete discharge)
E= 3125 Joules 

A LiPo pack with a cost of $20 is rated at 3.7V 4.4Ah for 1.0C and 0.2C to 3.0V

E= ½(Vi+Vf) *Ah 3600s/h = ½(3.7+3.0) 4.4Ah *3600 = 53,064 Joules [Watt-seconds]

You may compare the energy , cost and size with life expectancy then answer your own questions.