We know that in a critical system and for self-organized criticality we have long range interactions due to power-law decay of correlation. Is this fact equivalent to the butterfly effect?
No, or at least not in the sense the phrase "butterfly effect" is normally used. Well, possibly, but only if the critical system is chaotic.
The phrase is normally applied to systems that show chaotic behaviour. In such systems the trajectory of the system is very sensitive to the starting point i.e. if you take two points very close together in phase space the trajectories from those points will diverge exponentially fast. Hence the claim that a very small change to the system, such as a butterfly flapping its wings, will cause a big change in the future evolution of the system.
Critical systems may well show chaotic behaviour, and if so then I suppose the phrase "butterfly effect" does apply to them. However if this is the case the phrase would be used because they are chaotic not because they're critical.
There are similarities, but chaos and criticality are not equivalent.
Loosely speaking, chaotic behavior implies a fast loss of information, i.e., a shorter range correlation in time. In simpler terms, John Rennie's answer gives a definition of chaos and thus shows it's not equivalent to long range correlations.
On the other hand, if we consider the concrete example of a magnetic material, at criticality ($T\to T_c$: Curie temperature) the susceptibility diverges, which means its magnetization can vary wildly in response to small perturbations such as changes in the external field or thermal fluctuations. In this sense it's indeed reminiscent of chaos, but the dynamical systems concept truly related to it is bifurcation theory.