What do expressions even do?

Alt clicking gets you into an expression menu where you can link properties together or write code.

Left clicking is different. Left clicking mainly adds or removes keyframes.

Alt+click on stopwatch icon next to a property adds an expression.

Just click on stopwatch icon adds just a keyframe.

That's the difference.

Expressions allow you to extend what After Effects can do.

For example, I use expressions to create animated infographics where some values (like percentages for some statistics) animate from 0 to a specific value. You can't do that by hand, without expression. It would drive you crazy 😉

Not sure what you mean by the wick icon, but you can create a default expression by alt+clicking a property's animation stop watch icon. If you just click that icon it will create at keyframe at the current time (or remove all keyframes if one or more already exist).