I'd like to know whether the activation energy in Arrhenius equation is the enthalpy difference between the transition state and the reactant. If it's not enthalpy, is it Gibbs energy or something else?
The "E" is the height of the thermodynamic potential energy barrier. Which potential that is (enthalpy, Gibbs energy, Helmholtz, or internal energy) depends on the conditions under which the reaction takes place - which thermodynamic state variables are held constant.
For example, in a system where the number of moles, the pressure, and the temperature are held constant (called NPT), then the thermodynamic potential energy (and thus the activation energy) is measured using Gibbs free energy.
You can understand it as a difference in the potential energies of the reactants and the transition state on the potential energy surface of the reaction. Worth of note is the fact that the energy of activation should be treated more as an experimental parameter than a "real" value - for example you can write the Arrhenius equation for a complex reaction mechanistically composed of a few elementary ones - the E won't be a true barrier height then, more of a combination of the barriers for all elementary reactions.
Have a look at the Eyring equation as well, that one uses the Gibbs energy change.