First, easyCBM Lite only has NCTM (National Council of Teachers of Mathematics) math tests whereas the easyCBM Teacher Deluxe has both NCTM math and CCSS (Common Core State Standards) tests in both math and reading.
The math tests are all based on content standards. The CCSS math tests are aligned more closely with the Common Core State Standards while the NCTM math tests align with the standards for the National Council of Teachers of Mathematics. In addition, the CCSS math tests are a bit more challenging.
The common core standards are widely adopted across United States, and for those states that do not use common core, the national Council of teachers of mathematics standards are frequently adopted.
Both of these types of math tests are standards-based rather than computational/application based. They were all created to address the content standards as published by these two national bodies.
The CCSS in Mathematics have quite a bit in common with the NCTM Focal Point Standards which pre-date them by a few years.
Within easyCBM Lite Edition, the math tests that are available align to the NCTM Focal Point Standards. To the extent that the NCTM Focal Point Standards align well with the CCSS Math Standards at a given grade level, the easyCBM Lite Edition math assessments will provide useful information to teachers who are teaching based on Common Core.
Where CCSS and NCTM Focal Point Standards are not well-aligned, the CCSS Math Assessments available on the Deluxe Edition and District version of easyCBM would likely be more useful to a teacher who is teaching based on Common Core.
Another difference between the easyCBM NCTM-aligned and CCSS Math tests are that the NCTM measures are, by design, written to be more accessible and easier than the CCSS math measures. Thus, they are most appropriate for use with students who are really struggling and are performing below their grade level peers in the area of math. The CCSS Math tests may actually be a bit too challenging for students who are performing far below grade level expectations. In these situations, teachers may find it useful to use the NCTM-based measures until students make sufficient progress to perform at or above the 40th percentile before switching to the CCSS Math measures for regular progress monitoring.
In terms of which would be the better choice, it depends on what standards they align their curriculum to, and the skill level of the students being monitored. If they are teaching to the common core state standards, then the CCSS math test are more appropriate. However, for students who are really struggling, they may find the NCTM math tests more sensitive to monitoring growth.