This is already in some older threads. The "Republic of Texas" likely wouldn't survive long. It was a trainwreck IOTL, badly organized, feuding factions, incompetent govt, broke, its money worthless, and recognized by no one except France. (And the French ambassador spent his time in New Orleans getting drunk.) The one success it had was in ethnic cleansing of Indians, and to an extent, Mexicans. Only Mexico's own internal troubles kept it from getting returned to its mother country. And without a US invasion (which the issue of TX brought about anyway) Mexico's troubles drop off drastically. Chances are it's part of MX again within a decade or two.
If TX somehow survives past the first few decades, it's an economic backwater. Oil wasn't discovered IOTL til the 20th century. It might become a refuge for slaveowners if slavery is abolished in the US, and that'd contribute even more to its backwardness. It also makes it potentially the site of a Haiti-style slave revolt.