Being hard on yourself and being strict WITH yourself are 2 different things. To e hard on yourself is to not forgive yourself or going too far to punish yourself for mistakes. Being strict with yourself is about not allowing yourself things. It could be a restriction of food, or spending or smoking for example. If somebody says "Don't be strict with yourself" they are telling you to loosen up and take chances or to let yourself have fun.

"Don't be hard on yourself" is more common. As a native speaker in the US, I have never heard "don't be strict with yourself." "To be strict" usually describes a parent (or sometimes in the workplace, a boss) who is focused on rules. A more common sentence for this would be" "My parents were very strict." or "At work, my boss is very strict on us." "To be hard on yourself," means not forgiving yourself for mistakes.