I find this actually comical. I honestly don’t understand why anyone would compare UT Austin vs. Rice - they are totally different in every way. The SIZE and price alone are the biggest differences but in this case Rice will cost even less is a unbelievable. A rarity! In my opinion - you get what you pay for. You are getting a $60k+ education from Rice. That’s what most students pay to attend Rice (I believe over 70% are full pay). I can tell you without a doubt many UT students wish they could attend Rice. It is so incredibly hard to get into Rice. A family friend went to A&M and told my D if you can get into Rice, definitely attend. He attended A&M - chemical engineering. He said he was taught in huge auditoriums by TA’s and said that was a rip off. He ended up teaching himself and never attended classes. He only showed up for the exams! That’s terrible. He said he would never let his children go to a state university. State universities today are large because they rely on state funding and they accept as many as they can. Then they slowly weed them out as they lose their funding. Another family friend that attended UT Austin - petroleum engineer said EVERY CLASS was a weed out class. Rice doesn’t have weed out classes which is a HUGE difference.
My H, who attended UT Austin, even told me he had to go through 2 TA’s to get permission to see the professor. Once he received permission, he stood in line for 1 hour for only 5 minutes of the professors time. Yes, that was at UT Austin. My son never had to get permission and none of the SMU courses were taught by TA’s. One time my son fell asleep and missed his final and his professor called him asking him what happened… he told him to come to his office to take the exam. That was so kind and unbelievable. My H told me that would never happen at UT - he would have gotten a 0.
Why does it cost more? Its because it is a private university with smaller students and they don’t get funding from the state. They definitely offer the similar things that UT offers but with less students 3,000 vs. 50,000.
You should be comparing the total costs before aid - not after. You received a HUGE gift from Rice. Are you ready to take classes with 500+ students that is taught by TA’s (grad students in training) at UT? Rice gives you much more than UT would simply because you aren’t a number in a rat race. You will have practically everything available to you.
As far as jobs go, if you graduate in engineering you will definitely find a job. That isn’t an issue at either college. Engineering grads are highly sought after. Just so you know engineering, Architecture and medicine are the three hardest fields of study and they tend to have high drop out rates so you should have a back up plan in case you change majors. It is harder to major in anything at UT. You have to apply to the college after you’ve been accepted. So changing majors isn’t easy at UT. What that means is you can easily major in anything you’d like at Rice. You can also take any course you’d like. There aren’t rules where you may have to take every GEC course before you can start taking classes in your major (that’s a rule A&M). I don’t know if that happens at UT but the point is things aren’t as rigid at Rice.
So this brings up my last point - you need to pick the college that makes you the happiest. Happy students are successful students! My son chose SMU because he LOVES SMU (which has 5500 students). He thrived and is now working for the #1 software company in the world. I would attribute his success to SMU - 100%. His engineering professor lived in his dorm and they were fast friends. He became a TA - helping him grade his papers and helped out with tutoring and labs starting his sophomore year. He took 4-5 engineering courses freshman year! He was offered the world - TA, co-ops, internship, research, masters, etc. Its all because of the small tight knit at SMU.
In summary, my advice is you should always pick the college that fits you. None of us know you. Only you do. My vote is Rice but if it doesn’t fit you then it doesn’t matter. If you aren’t happy, you won’t thrive at Rice. If you will be happy being part of UT, then go for it. The Honors program will help but you’ll still have a lot of barriers. You’ll just have to go through it just like the rest but being happy is definitely key.
Just a final note, my H said he would attend UT all over again even though he was treated like a number. He said it was a huge wake up call and that alone was a life lesson. He’s nuts! ha ha. However, he said he wished he could have attended SMU - it way too expensive for him at the time and he’s proud our son graduated from SMU. The ironic thing is now my H is part of SMU Grad school - engineering. Funny! Also, my D is at WashU which is very similar to Rice. She LOVES WashU. When the kids/family are happy, mom is definitely happy! 
Good luck with your decision!!