Continuing an action - てくる vs つづける

I’m not sure I really understand the difference between these two grammar points when they mean to continue an action. It seems that てくる means more to “come to” a state and persist in it and つづける means to continue a voluntary physical action but there still seems to be a lot of overlap and so I get reviews wrong.

For example, both of these are examples for their corresponding grammar point.

今日の天気は晴れのはずだったのに、雨が降り続けますね。
It was supposed to be a clear day today, but it continues to rain.

お金がなくなってきた。
I have continued to lose money.

To me the nuance for both of these grammar points are mostly the same. Both of the states have “come to pass” and result in a continuing state. What’s the difference? This sort of confusion has be missing reviews like the one below: