Fidya should be given in Ramadan. Allah the Almighty commands:
“For those who can endure to fast (in Ramadan), a fee to feed a desperate person is needed.” (2:184)
The Messenger of Allah said:
“Whoever gives the charity of Ramadan before the feast prayer, it is valid. Whoever gives it after the prayer, it is just a kind of charity.” (Abu Dawud, Zakat, 18)
Zakat on the other hand, is an annual tax. It is given from all of the possessed money, gold, animals and commodities in the end of the year. It can also be given monthly by calculating the annual income and dividing it to 12 after subtracting the annual costs and debts. Zakat is not related to Ramadan.