So does UT.
Here’s the thing:
The name matters… Cockrell is a top name- don’t let familiarity breed contempt
The opps- both will have more than you will have time to take advantage of
The contacts…if you want to go into research, then the contacts you need are people who are respected in that field. Both places will have all more than enough contacts for you- and your peers at both will be other high achievers who will spread out across the country and become your contacts and peer group.
IOW- you won’t be getting anything less by choosing the one that will cost you less money, and you will be getting more by getting the major you want.