Texas left Mexico in 1836, and seceded from the Union in 1861, so I imagine both these movements involved some of the same people. Did they think the Union had become oppressive? Were they wary of joining yet another country? Did they want to become independent once again? Also, did Lincoln treat Texas specially because both movements involved the same people?
The then governor, a former General Sam Houston of the battle of San Jacinto fame, refused to support the Confederacy and was removed from office. While he was in favor of states' rights, he believed that the North would win and it would only lead to death and destruction for the South. Lincoln offered to lend him federal troops to keep Texas in the Union but he declined as he wanted to avoid bloodshed.
I hope this helps.