In most ways these two constitutions were very similar. Most of the Confederate document was taken directly (often verbatim) from the Constitution of the United States. This is not surprising given that the Confederates generally disagreed with the Northerners on what the Constitution meant, not on whether it was a good document.
The main differences between the two are meant to emphasize what the Southerners thought the Northerners didn't understand. For example, there is a greater emphasis on states' rights. The CSA's preamble explicitly states that the states are "sovereign and independent" and that the states are the ones forming the national government. The other major difference has, of course, to do with slavery. Here, there are many more specific protections for slavery than there are in the US Constitution.