I watched both movies recently and along with some similarities and differences other than the oil theme, both portray elements of the human character and contemporary society exceedingly well through stellar acting performances. The subtle similarities that I found are : upward social mobility, tension between an old and new way of life, and eventual overlap of social issues. Differences: racial undercurrents (GIANT) vs. religion (TWBB), and the land - oil struggle (GIANT) vs. Big oil-independent prospector struggle (TWBB).
Speaking to the similarities, oil was an obvious means of upward movement, exemplified by James Dean's and Daniel Day Lewis' characters. In GIANT, the tension in the changing Texas economy with ranching community on the one hand and oil tycoons on the other was also demonstrated. In TWBB, the tension was revealed more in DD Lewis' role, where he continued to sleep on the floor and live like a wildcatter in spite of his riches.
The racial undercurrents in the GIANT highlight the immigrants' struggle that is widely experienced in Texas even today. This is not completely isolated from the population's religious beliefs, which are dwelt upon more in TWBB.
As an aside, I noticed that people still ask the same questions about Texas today as they did half a century ago. I was traveling in New England recently had to explain, at length, how about the size of and peoples' occupations in the Lone Star state.
Another aside, I think that movies are a great way to instill an awareness of energy issues in the minds of the general populace.