It seems to me that the chief difference between the MIT license and GPL is that the MIT doesn't require modifications be open sourced whereas the GPL does.
True - in general. You don't have to open-source your changes if you're using GPL. You could modify it and use it for your own purpose as long as you're not distributing it. BUT... if you DO distribute it, then your entire project that is using the GPL code also becomes GPL automatically. Which means, it must be open-sourced, and the recipient gets all the same rights as you - meaning, they can turn around and distribute it, modify it, sell it, etc. And that would include your proprietary code which would then no longer be proprietary - it becomes open source.
The difference with MIT is that even if you actually distribute your proprietary code that is using the MIT licensed code, you do not have to make the code open source. You can distribute it as a closed app where the code is encrypted or is a binary. Including the MIT-licensed code can be encrypted, as long as it carries the MIT license notice.
is the GPL is more restrictive than the MIT license?
Yes, very much so.