What's the difference between a breast lump and a cyst?

Less/more specific: A lump can be anything - solid or fluid filled, scar tissue, just a prominent area. It is a very non-specific term. A cyst means a fluid filled structure.

Solid vs fluid: A cyst is full of fluid, a lump is generally solid. Both need to be investigated carefully and probably biopsied.

Solid v Fluid-Filled: The most common causes of breast lumps include "lumpy-bumpy" breast tissue ("fibrocystic changes"), tumors (benign or malignant), and cysts (fluid-filled;benign). A thorough history and physical examination supplemented by ultrasound+/-mammography+/-biopsy will help sort this out. Your gynecologist or primary care doctor can help you sort this out. Good luck.