To put it simply: compute engine gives you a server which you have full control/responsibility for. You have direct access to the operating system, and you install all the software that you want, which is usually a web server, database, etc...
In app engine you don't manage the operating system of any of the underlying software. You only upload code (Java, PHP, Python, or Go) and voila - it just runs...
App engine saves tons of headache, especially for inexperienced people but it has 2 significant drawbacks: 1. more expensive (but it does have a free quota which compute engine doesn't) 2. you have less control, thus certain things are just not possible, or only possible in one specific way (for example saving and writing files).