What is a difference between a license and a copyright contract? It seems that they achieve the same thing. So what are the use cases for both?
This answer is limited to United States law. The situation in other countries is definitely different.
Under United States law, the owner of a lawfully made copy of a copyrighted work has, as a right of the physical possession of that work, the right to the work's ordinary use. Licenses grant you additional rights such as the right to make derivative works, the right to make copies beyond what's needed for ordinary use, and so on.
A pure license doesn't ask for anything in return, it just gives you new rights. Those rights may be conditional, but the conditions are just things you have to do to get new rights. They're not conditions imposed on any existing rights you had.
A contract is an agreement between two parties. Both parties must agree to a contract for the contract to be enforceable. Contracts can take away rights you otherwise have.
You don't need a license or contract to use a copyrighted work if you lawfully possess a copy of that work. Say you download a copy of a work covered by the GPL. You can refuse to accept the GPL license and you can still use that work. Why? Because no law prohibits you from doing so and there is no civil cause of action for using a lawful copy of a work under US law.
But now say you want to give a copy of that work to your friend. This is illegal under US law because 17 USC 106 restrict that right to the copyright holder and there's no applicable exception since that's not part of the ordinary use. For a work covered by the GPL, the license offers to give you that right, a right you wouldn't otherwise have. It imposes conditions on you that are specifically in exchange for the grant of the new right.
If you do accept the GPL, it functions as a contract. You got in exchange a right you didn't have before and if you don't comply with the GPL's terms, you don't have the copyright holder's permission to exercise those rights which is required under the law.
By contrast, a click-through or EULA takes effect when you agree to it and such agreement is a condition for using the software. That's a pure contract and usually doesn't give you any rights you wouldn't have in the absence of such an agreement other than the use of the software.
Many copyright licenses are also contracts. Most copyright contracts will be, or will include, licenses. Free or permissive licenses, such as the CC-BY license, might not be considered contracts though lack of consideration, although in the case of the CC-BY license, the promise to provide attribution might be sufficient consideration. Something like the CC-Zero license, where the copyright holder basically says "anyone may use this with no conditions" would probably not be a contract in any jurisdiction that requires consideration for a valid contract.
The use case is that when the copyright holder wants something back from a license, such as a fee or attribution, the holder should use a license that is also a contract. When the holder is not granting any use rights to the copyrighted work, a contract will not include a license, which is precisely a grant of such rights.