There is no big difference both of the fields deal with computer and software, computer science works on the theories behind how the computer works while software engineering works on developing new applications depending on the requirements.

The Difference:

Computer Science deals with studying and analyzing the algorithms and problems that are related to making the computer do the task. This field focuses more on how these computer programming languages work.

Software Engineering deals with creating new software. In this field you require to actually work with a team so that you can build a new application. Software Engineering focuses on how to design & build the Software.

Good Luck!