Quoting the great Wikipedia article on ABC (emphasis added):
Approximate Bayesian computation (ABC) constitutes a class of
computational methods rooted in Bayesian statistics. In all
model-based statistical inference, the likelihood function is of
central importance, since it expresses the probability of the observed
data under a particular statistical model, and thus quantifies the
support data lend to particular values of parameters and to choices
among different models. For simple models, an analytical formula for
the likelihood function can typically be derived. However, for more
complex models, an analytical formula might be elusive or the
likelihood function might be computationally very costly to evaluate.
ABC methods bypass the evaluation of the likelihood function. (...)
When using ABC we approximate the likelihood function using some kind of summary statistics, so you rather would not use it instead of non-approximate Bayesian computation. We can use it in situations, where we cannot use the non-approximate methods.
Check the famous socks example by Rasmus Bååth for some friendly introduction (see also here).