"Homework" is the general category and without "a piece of..." is not a countable noun, like the rest. For me, task is just a translation of tarea, but ever used with homework in the UK, more for a household task or chore (or like a tramite en español, por ejemplo). Assignment, to me, is more important than simple homework.

In this case, no, there is no difference between them and they could all be used to label work you need to do at home for a class. Task and assignment are nearly always interchangeable; they are synonyms for one another. But 'homework' really refers only to work related to school. If you are talking about work you do at home for your actual profession, for example, you would not say that you are doing "homework."