Categories are a bit like budgets. You might start to notice how some things don't need a budget, but do need some kind of meta-thing. A category might work. "Furniture", "interest", "shoes" and "lunch" are perfect categories.
But there is also income (deposits) that you might want to give a category:
Each transaction can be given a category. This will show you what you spent your money on, or where it is coming from. Firefly III allows you to dynamically create and manage categories. Fancy charts will show you how your money is divided over categories. There is a category report that can show you exactly what is happening within a category.
If you try to save money every month on a certain subject, it's a budget. Groceries are budget. Bills are a budget. If you travel by train occasionally, it's not a budget.
First and foremost: a category is "incidental". You don't buy new furniture every month but you might want to keep track of such expenses. Or you don't care about costs for public traffic (budget-wise) but a category would be nice.
The rule of thumb is: would you make a real life envelope for it? If yes: budget. If no: category.
Categories can be used in deposits (earning money). Budgets can't.