Illustrator makes vector images ( when you make a mark, the computer creates a mathematical equation for that mark, therefore using it is not as intuitive) Photoshop creates raster images, which means it works with pixels, which makes it so you can draw more like you draw in real life, and as the name suggests, work better with photos. If you already knew about vector and raster images, i'm sorry. I just learned about them myself. Illustrator takes longer to get used to and get good at, and mostly works for making entirely new images. It's also fun to use the two in conjunction (i.e. make some text in illustrator, since you can manipulate it really easily, then use it in a Ps image.)They do very different things, and if you are going to use both, you should have both. I just downloaded the 30 day trial and it didn't cost me anything, so if you don't think you will use it after your course, that's an option. Also if you're a student you can usually get a discount through your school's bookstore.
Picture I made in Photoshop using illustrator for the text parts (bottom right) I've only used it a handful of times, so judge accordingly:
Don't mind me.