I've worked 3 projects as a contractor for Exxon, and now have been a Chevron employee for the last 4 years.
The two companies are completely different in attitude. At Exxon, they almost expect you to work an extra 10-12 hours per week. 40 hour weeks do not exist. Here at Chevron, people show up late and leave early every day, and nobody seems to care about that. They also do not have the 9/80 schedule, it is 5 long days per week - usually 10-12 hrs a day.
Pay is comparable between the two, until you factor in the extra hours at Exxon.
Casual Fridays at Exxon consist of dress slacks and button up shirts with a collar. No jeans, no sneakers, no t-shirts. It is a very old fashioned company, too many stiffs work there. Overall, I find the people at Chevron easier to work with, and have more outgoing personalities and leadership characteristics.
At Exxon, you have to wait your turn to be a leader, they will smack you down if you try to step up too early. You just do your job there. That being said, I would rate the management at Exxon as much better than Chevron.
They promote managers who are experts in their field, and are proven top-level performers. They penalize these managers when they are wrong. At Chevron, I've mostly found incompetent managers who just have good personalities. They can blow millions in change orders, and still keep their job.
That is a problem, there needs to be a balance of both of these situations. Chevron is horrible at demoting or firing managers, which is a shame.