Most of the Software Engineering is Computer Science. Lots of Computer Science is not Software Engineering.
Given the cost of education (I assume the USA), I would go for CS as it is more fundamental and long-lasting knowledge. Software Engineering is more hands-on and more sensitive to changes in technology.
Hypothetically, Software Engineering makes you slightly more “job ready”. In reality – if you write code and enjoy doing it, you are hire-able with either degree. And without one, too. Especially if you are able and willing to keep learning.
Don’t overthink it too much. You need to develop software to be a software engineer. “Coders gonna code”.
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