Like J. Ríos said, they're not always interchangeable. For example, you could say, "I will buy eggs on the way home." But you can't use 'manner' in this context. This usually happens when "way" refers to a route or path.

In some cases they are interchangeable. It is the exception not the rule..

It depends on context, "I like your manner / I like your way." ,but not always.

They are usually interchangeable but ‘way’ can hint more towards the process and steps you may have taken whereas ‘manner’ implies more the style of your actions. ‘Way’ tends to be used more often as it sounds less formal than manner but you wouldn’t be wrong to use either!