I've always heard from both UT & A&M administrators, students, and alumni that THEIR school is the oldest in Texas. As a graduate of TAMU, I know 1876 is an important year (on the seal/ring) and that's about it. Meanwhile, over at BON (Burnt Orange Nation) they like to throw around a quote from the Texas Constitution, something along the lines of A&M is the Agricultural and Mechanical branch of UT, the flagship University in Texas. This confusion led me to do some research. I found a great synopsis on the Texas Almanac LINK.
The source sums it up nicely: "The two major university systems in Texas had slow and shaky beginnings."
1839 - The Congress of the Republic of Texas set aside a 40-acre site (ahhhhhh finally makes sense) for the construction of a university, but no plans were made. Additional land was also set aside in order to endow two universities.
1846 - Classes begin at Baylor, acknowledged as the State's oldest continually operating institution of higher learning.
1858 - A bill establishing the University of Texas passed the Legislature. Progress was delayed on account of the Civil War.
1871 - A bill passed by the US Congress in 1862 granted Texas 180,000 acres to establish an agricultural and mechanical college. After delays things got rolling in 1871. After considering other sites, Bryan, TX was selected.
1876 - Construction of the A&M campus began in 1874 and the A&M College formally opened on Oct. 4, 1876.
1876 - The Texas Constitution (1876) provided that a "university of the first class" be established and it be called "The University of Texas." The A&M College was to be a branch of the main university.
1883 - After site selections and construction, The University of Texas formally opens Sept. 15, 1883.
What did we learn? UT was the first of the two to be established. However, A&M was already running classes before UT had even picked a site. While the Constitution provides that A&M be a branch of UT, the two have always had separate Board of Regents and have never acted together. Thus, I'm prepared to conclude that A&M is the oldest University of the two and was never realistically intended to nor has ever acted as a "branch" of UT.