Yes, there is a difference. A child is anyone younger than 18, though some use it for people older than that. Typically, below the age of 13 we use child and for 13 to 19 we use teenager. There are some new terms like tween, tweener and pre-teen, but these are all more slang than regular everyday words. A kid is a bit less clear. While it originally was used as a nickname for child, it often can be used for any person that the speaker thinks is considerably younger than they are. So, I happen to be 53 years old. I could call a 30 year old a kid. While someone that is 75 could call me a kid. While the older person could also use child in these examples, kid is more common. Finally, kid is a nickname used frequently for men in the last 20 or so years. In this usage, playfullness, energy and curiosity seem to be implied rather than a specific age.

The word "kid" is informal compared to "child". Someone might be called a kid at any age, but not a child (unless receiving a whiplash from a spouse). There's no age formula.