How do I keep my cool?
One way to do this is you can tell someone (not necessarily the exec committee, maybe whoever greets you at the door) that you have nerves. Also, take a pause before answering, which not only gives time to collect thoughts, but also overcomes the "nervously rushed answer".
How might an interview with an executive selection committee be different than a "regular" interview?
I don't know how senior your position is, but most line manager and below interviews will have different people asking different types of questions.
Example for an engineering interview: Committee made up of 1 HR person, 1 technical SME, and people manager. HR person may be there to ask generic questions, technical SME will test you with design scenarios, and the people manager will likely ask behavioural / culture fit questions.
A couple of tips from Ramit Sethi:
-Bring something with you of value (his Briefcase Technique): it's a great piece of showmanship if you get asked a question, and are able to go "actually, I created something similar recently" then pull it out your folder and hand it to them (be wary of confidentiality of course)
-Fully support lbmustache's good questions tip: what is the biggest pain that this role must solve in the next 3 months? What is the office culture like? How is the role / team's performance measured? The closer you can make it sound like you're pretty much already working there, the better.