Follow-up question to this: Was there any particular Texas Republic laws on the books that clashed with ones from the United States or could be considered flat out strange to the rest of what the Union's citizens were accustomed to at the time?
UT-Austin has a good portal on the Texas Constitution. From the preamble:
We, the People of Texas, in order to form a Government, establish justice, ensure domestic tranquility, provide for the common defence and general welfare; and to secure the blessings of liberty to ourselves, and our posterity, do ordain and establish this Constitution.
You can tell that the Republic's founders had the broader American example in mind while drafting their constitution. Indeed, like the U.S. Constitution's Article I, Texas's Article II established a government of enumerated powers but as distinct from the United States, Texas was not a federal republic; it didn't have "states" or other subsidiary entities.
Some other points of distinction are, the 1836 Texas Constitution explicitly mentioned and continued slavery while also enacting an "equal protection clause"; and forbade ministers from serving in public office.