Simply put, because A LEFT JOIN B can produce different results from A RIGHT JOIN B. Whether or not those two join types produce the same result mostly depends what you want to see happen if matching records don't exist in one table or the other.
Let's say you have a table for customers, and each customer can have several addresses, so you have another table for addresses. If you use a LEFT join between them, you'll get a row back for the customer even if the customer doesn't have an address at all. If you use RIGHT join, you'll get a row back for every address that fits the WHERE condition, whether the customer record exists or not.
But if matching-ID records exist in both tables, then as you've probably observed, you'll probably see identical results regardless of which table is on the left or right - depending on what's in your ON and WHERE clauses, of course.
There isn't likely to be a noticeable performance benefit (or degradation) in using B RIGHT JOIN A as opposed to A LEFT JOIN B, as long as both tables are indexed on the join fields. That's a huge generalization, though - every database is different, so you'd just have to try it both ways and choose if you want to optimize things.