You can hand off a problem to someone else in prayer, even though the other person (God) does nothing about your problem, you still feel better because it's no longer your problem.
When you believe God is real, it also has the effect of talking to someone else who you feel understands your problems, which results in a positive emotion.
Furthermore it helps to focus your mind on something, like meditation, which can have a positive effect and can also cement something into your subconscious mind so it will resurface later. Like praying fervently to stop eating cookies, and then when you want to stuff one in your mouth, you think "oh yeah.. no more cookies today."
It can also be a planning tool by telling God what you did today and what you plan to do tomorrow.
It can help you sleep because you clear your mind and get mentally ready for bed.
It can help you gain the confidence to tackle a problem because you feel that someone is supporting you.
It can heighten your appreciation of something like food because you are more focused and thankful than when you just start eating.
Summarizing, it can be very helpful and God is not required, but in the end you are only fooling yourself.