They are related ideas, but an asymptotically unbiased estimator doesn't have to be consistent.
For example, imagine an i.i.d. sample of size $n$ ($X_1, X_2, ..., X_n$) from some distribution with mean $\mu$ and variance $\sigma^2$. As an estimator of $\mu$ consider $T = X_1 + 1/n$.
(Edit: Note the $X_1$ there, not $\bar{X}$)
The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.