A Texoma license allows you to fish anywhere on Lake Texas - both Texas and Oklahoma water.
A Texas license can be used to fish only parts of Texoma that are in Texas and the same for an Oklahoma license. Good luck knowing where you are at.
Having both Texas and Oklahoma licenses allow you to fish anywhere in Texoma. This is often a better choice than purchasing a Texoma license if you plan to fish Texoma as well as wanting to fish other Texas and/or Oklahoma waters. This is what I do.
If you are an OK resident and 65 or older, you don't need any license to fish Texas waters (but you may need ID showing age).
If you are a Tx resident and 65 or older, you need a Tx license but it can be a regular license or a senior citizen license (reduced price) as Lena said above. If you are a Tx resident and born BEFORE January 1, 1931, you are exempt from any licensing requirement (again, proof of age may be required and don't drool on the GW when you show it to him).
There may be other nuances in the state laws but I think this covers most situations. As always - don't just taken internet postings as the gospel - always check them against what each state has in their regulation handbook - all of us are liable to make mistakes.