Hello!
Independence: Texas, as you know, was a Mexican State before 1835-1836 and before those times. A vast influx of Americans, specifically Southern Americans, came through the border, even after it was declared Illegal to cross. They brought with them American values, now it was required that a settler coming into Texas become Catholic in order to settle down. I imagine that some just agreed but remained Protestant by heart.
They brought slavery, which was eventually outlawed, leading to revolution, but before, or during, these factors, Mexico's government shifted and I believe there was a revolution of sorts carried out by Santa Anna, the Cat with Nine Lives, who promised Texans, along with his Mexican peers, rights that they didn't have under the previous government.
Well, Santa Anna became Dictator, which angered Texans along with Mexicans, and he probably did another assortment of things to anger them all, including the ban on slaves towards Texans. War happened, Texas won, now to the present day.
Texas, under independence, had its own laws, its own money, its own Presidents, and a effective military competency to defend their lands from Comanche Indians and the Mexican Army, which lost another big battle after Texas Independence. Texas was a well-established country of its own, its culture today reflects that, so why become apart of the US?
Maybe it was fear of being taken over, unlikely, or the security of being in the Union with the other states. Perhaps, Texas shared such an identity with the other states that they joined out of the latter reason, security.
Statehood: Mexico was gunning to take-over Texas once again, even after it was signed away by Santa Anna after San Jacinto, and Comanche Indians pressed in from Northern Texas. If Texas joined the Union, then they would receive benefits accompanied by such an act, and most importantly: they would have the security in their economy and safety of their borders with their two enemies.
More Americans, sharing the same identity with Texans, would flood over the border, making the largest state in the Union, at that time, into a boom-state where land was gobbled up and cities were erected from nowhere. However, despite the good of more land for both parties, Congress was hesitant to accept Texas's Constitution out of fear for up-setting the slavery balance between North and South, which was an on-going issue since the Revolution.
It took some time, but it happened, and in annexing Texas, Mexico declared war over a boundary dispute between the Brazos, Mexican definition of Texas's border, and Rio Grande, America's boundary for the state.
There you have it, hopefully I answered your questions.
Cheers!