Tearful/Weepy

They're very similar in meaning - though weepy is perhaps a bit stronger : I'd understand the person was more upset if they were "weepy", and probably depressed - while a child might be "tearful" one moment and fine again the next. They are often used interchangeably though. Here are some examples from the Cobuild concordancer:

... however, Nigel became depressed and tearful. Eventually he became angry with
... swings and they may become depressed, sad and weepy.

I don't see much difference here.

But of course, they are different in the sense that they often collocate (= go together with) other words, and may not then be interchangeable. For example, you can talk about a weepy film (ie a sentimental film that makes you cry) or a weepy love-song but not about *a tearful film/song

Interestingly the concordancer had a lot more examples of tearful than weepy - it's clearly more common.