I am trying to to get an intuitive understanding and feel for the difference and practical difference between the term consistent and asymptotically unbiased. I know their mathematical/statistical definitions, but I'm looking for something intuitive. To me, looking at their individual definitions, they almost seem to be the same thing. I realize the difference must be subtle but I just don't see it. I'm try to visualize the differences, but just can't. Can anyone help?
They are similar; a consistent estimator that's biased must nevertheless be asymptotically unbiased (otherwise it could not be consistent), 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 $T = X_1 + 1/n$.
The bias is $1/n$ so $T$ is asymptotically unbiased, but it is not consistent.
There are "unbiased but not consistent" estimators as well as "biased but consistent" estimators:
So, they are not the same thing.
Also, there is a long discussion about this topic here:
Asymptotic unbiased: As $n \rightarrow \infty$, bias converges to $0$.
Consistent: As $n \rightarrow \infty$, variance of the estimator converges to $0$.
I would like to clarify that consistency in general does not imply asymptotic unbiasedness. Consider an estimator for $0$ taking value $0$ with probability $n/(n-1)$ and value $n$ with probability $1/n$. It is a biased estimator since the expected value is always equal to $1$ and the bias does not disappear even if $n\to\infty$. However, it is a consistent estimator since it converges to $0$ in probability as $n\to\infty$.
Asymptotic unbiasedness does not imply consistency either as it is mentioned in other answers. For example, the periodogram is an asymptotically unbiased estimator of the spectral density, but it is not consistent.
Roughly speaking, consistency means that for large values of $n$ we are going to be close to the true value of the parameter with a high probability, i.e. estimates are going to be close to the true value of the parameter. Asymptotic unbiasedness means that for large values of $n$ on average we are going to be close to the true value of the parameter, i.e. the average of estimates is going to be close to the true value of the parameter, but not necessarily the estimates themselves.