1845: Texas doesn't feel like nominating itself for Statehood since its a good summer what would it look like today aside from having George W. Bush as its most likely President.
It would have a very hard time surviving to make it to post-1900. First, Mexico might have decided to reclaim it anytime. Second, it was foundering in debt it couldn't pay -- those debts were a huge reason it became a state.
Norton is correct that a foreign protector might address these problems somewhat, but there would be a price in accepting one.
The US will likely still find reasons for conflict with Mexico and will likely wind up with Mexico. The US and TX might have some border wars over what is now AZ, NM, CO, etc. TX will probably be beaten down to something like its current size.
Assuming the US still has a civil war (though TX's absence shifts the political balance somewhat), TX might choose to support the CSA. However, their support as a foreign nation would be less than their support as a CSA state. CSA still loses.
Anyway, if TX survives all that and doesn't decide to become a state later, it'll be a small country between two larger countries. It will always be economically and politically tiny relative to the US. It will try to avoid pissing off the US too much. It may eventually develop a relationship with the US like the one the US has with Canada.