Think of 珍惜 as treasure (verb) and 爱惜 as love (verb). You can treasure something without love or vice-versa. 我珍惜你的才華但曾未愛惜過你的心.

珍惜is to cherish something precious, such as the time。 爱惜 means not being able to destroy what you like

My understanding is like 珍惜 is more likely for something abstract. For example... 珍惜時光（time) or 珍惜相聚（réunion). For 愛惜, it’s for something concrete, ex... 愛惜身體（body) or 愛惜東西（object). I think they aren’t quite different, and it’s easy to understand and nothing wrong if you don’t use these words like what I mentioned above.

我自己的理解，“珍惜”更靠近英文中的treasure sth.，比如珍惜某样昂贵或很有意义的物品。可以是“珍惜时间”，“珍惜和他的友谊”，“珍惜这块手表”之类的。而“爱惜”更强调保护某样东西不受伤害(take care of sth. or someone)。比如“爱惜衣服(不想让衣服损坏)”，“爱惜眼睛(不想让眼睛疲劳，难受等)”。