I am confused about this. Are they the same? There are some problems in which I am asked to find the Uniformly Minimum Variance Unbiased Estimator (UMVUE) first and then check if it achieves the CRLB. Yet from what I've read, I understand that an unbiased estimator that achieves CRLB is UMVUE. Thanks.
They are not the same. Achieving the CR lower bound is a sufficient condition for an unbiased estimator to be UMVUE. However, it is not necessary.
For example, Example 3.10 in this link gives an estimator that is UMVUE (by the Lehmann-Scheffe theorem) but does not attain the CR Lower bound.