First, I am wondering to know why you consider them to be phrases, not sentences.

Hi In fact the second sentence is slightly incorrect. You have two different grammatical forms. You can just say "I am wondering if..." because 'to wonder' is a verb . And the form is Present Continuous. It doesn't require 'to know'. The other sentence needs the verb 'to know' because 'curious' an adjective. "Curious" is a state not an action. So in this case you do need 'to know'. And that sort of answers your question about the meaning. 'curious to know' has the same sort of meaning as 'to wonder' in most cases. ("I am wondering why you are studying English." = "I am curious to know why you are studying English.) However, like many words they have different forms and meanings in different contexts. I won't confuse things too much, but "Curious!" as an expression normally means "Strange!", and "I wonder" sometimes expresses strong doubt. ("I wonder if he will buy the drinks." = "I don't think he will pay for the drinks.")

I am curious to know. - ok I am wondering to know. Nope. NOt like this I am wondering <about something> I am wondering whether this is clear. I am curious to know whether this is clear. I am wondering about Harry. Is he ok? I am wondering whether it will rain tomorrow. I wonder whether it will rain tomorrow. I wondering what the weather will be like tomorrow. I am curious to know what the weather will be like tomorrow.