The Mexican seal was probably the basis for the live oak and olive wreaths that have made their way onto Texas' present seal. In 1845, when Texas gained statehood in the Union, the images of the Republic's seal were retained, and the word "Republic" was replaced with "State".
The olive and the live oak branches can be considered to represent peace and strength respectively.
The flag was adopted as the state flag when Texas became the 28th state in 1845. As with the flag of the United States, the blue stands for loyalty, the white represents strength, and the red is for bravery.