Software Engineering falls under the umbrella term of Computer Science. Software Engineering degrees usually provides you with less knowledge in other areas, while CS will provide you a wider aspect, including AI, Machine Learning, Databases, Big Data, Data mining, and maybe also things like, Consultancy, System Analysis, Maths, other IT theory modules etc. The modules are depending on the university though.
While Software Engineering is focusing on Software engineering principles, HCI (UX,UI), Data Structures and Algorithm, Advanced Programming, Software assurance and testing.
Then you have modules that are usually provided at both courses: at least 1 semester of Maths, Project management, website development, mobile app development, programming fundamentals, maybe enterprise systems, and some databases at least a bit. (more on CS in my opinion)
Going through these modules, you can see how these courses differ. Each of these courses usually have at least 1 programming module per semester, so your knowledge in programming would be very similar. (Software engineering takes it only 1 step ahead, which is an extra module in the 3rd year named advanced programming, that's quite usual, and also spends more time on SDLC in most modules)
If you're interesting in game development, that's a completely different route. Game development & Digital Media degrees are quite aliens compared to other IT related courses in my opinion. (At most places, you can't even do a CS masters with gaming background, how cruel!)
They usually only do games and games, and have no other knowledge like maths, databases, but they also do at least one module in web design and and mobile app development as well. I only had a couple mutual lecturers with game students. (project management, web and mobile apps)
CS and SE degrees don't usually include any game dev modules, but hey! You'll have a degree, that means you'll be able to specialise yourself into any area you like. Also, having a CS degree will open more doors for masters, but also for work. CS graduates usually end up at the same place as the software engineers, but they had to concentrate to a bit more area whilst their study. In return, as a software engineer graduate, you won't really end up doing researches in AI/ML at PhD level, or work with databases, or do things that is usually more theory than coding.
So each has its own advantages and disadvantages. If you don't know what area you want to work in, I would suggest you Computer Science. Also CS, if you want to have a wider knowledge in IT and want to have a bit of a taste of everything. SE degree is for you, if you're 100% you'll be a programmer and don't want to waste time on other things that are not engineering.
You don't choose the course, you choose the modules. At some places, there are no difference between these courses other than the name, but there are other places where you have so much difference that you have almost nothing in common. (Therefore, less opportunities in other areas but also for masters, as some MSc course requires you to have a wider knowledge and completed general CS modules - I heard that on a PG open day)