Difference between criterion validity and reliability equivalence In the book Doing research in the real world by David E. Gray, there is a section on experiment design. When discussing validity and reliability, the author defines “criterion validity” as 


  This is where we compare how people have answered a new measure of a concept, with existing, widely accepted measures of a concept.


and a little later, in the “Reliability” part, there is a subheading “Equivalence”, which says 


  Another way of testing the reliability of an instrument is by comparing the
  responses of a set of subjects with responses made by the same set of subjects on another instrument (preferably on the same day). 


So if I got this right, we are both times measuring if there is a difference between the answer on our new instrument and another, existing instrument. 

Is there a practical difference between the two concepts, or only a philosophical one? And whether practical or philosophical, what is the actual difference? 

Update The author discusses validity and reliability in general, then lists 7 different aspects of validity and 5 different aspects of reliability. "Criterion validity" and "reliability equivalence" are only one type of each, respectively. Please consider in your answer that this question is not about validity vs. reliability in general, but only about these two specific aspects.
Criterion validity concerns with measuring the right thing. For instance, GPA is likely to have criterion validity to measure a student's academic understanding. While the change in weight in the last semester has much less criterion validity to measure the same trait. Basically, if the measurement you use and the trait you want to measure has a high correlation, then there is likely criterion validity.

Reliability concerns mostly with measuring the thing right. For instance, if GPA can measure a student's academic understanding, and percent attendance can also measure a student's academic understanding, then GPA and percent attendance should correlate, aka, they are reliable. Before subjected to reliability assessment, the tests are usually checked if they are criterion-valid. However, it's possible to have two tests that are highly correlated (reliable) but invalid. Such as using dietary fat intake and serum lipid to predict a college graduate's earning potential.

Notice that there a few different types of reliabilities, the one you cited is more about alternate forms reliability, there are also test-retest reliability and inter-rater reliability, etc. 

Practically, they are not interchangeable. Validity happens between the true trait (or behavior) and the measurements. Reliability happens between two measurements (or modes/instances of measurement.)