I am interested to know if, for some, there is a subtle difference between the two phrases in the title. I am equally interested in knowing if there is a subtle difference.
There is definitely a difference to me. They do not mean the same thing.
Being interested in something describes a general interest that you hold—something that you tend to find interesting and devote some part of your time doing or at least thinking about.
Being interested to do something, on the other hand, implies that right at this moment, you find the thing in question interesting and would like to [do whatever it is]; but it does not speak to your general interests.
With the verb know, the latter tends to be a more accurate description of what’s going on: knowing is not something that we often need to talk about as one of our interests in life. For me personally, I frequently use it in the conditional:
I’d be interested to know how much of our recycled waste is actually recycled.
In other words, it just struck me now that that particular piece of information would be an interesting one to have; I don’t count knowing how much of our waste is recycled among my general interests, though.
If I do count something as a general interest, I would use the gerund form:
I’m interested in knowing everything I can find out about the mating rituals of grasshoppers.
This implies two things:
That one of my general interests is the mating rituals of grasshoppers, and I therefore spend lots of time in learning everything I can about them (similar to “I’m interested in art”, except with an action as the interest instead of just a broad concept).
I have very little to no social life.
This is not really a hard-and-fast distinction, but it’s a good rule of thumb. There are many cases where a gerund construction can, from context, be shoehorned into the sense of ‘current, fleeting interest’ rather than ‘permanent, general interest’:
Would you be interested in going out for dinner tonight?
Would you be interested in cooking for me tonight?
This is partly to do with the conditional, which tends to reinforce the ‘currently’ sense of interested; if you describe general interests, you would usually just use a straightforward present:
Are you interested in going out to restaurants?
Are you interested in cooking?
These last two are definitely ungrammatical to me with the infinitive construction:
*Are you interested to go out to restaurants?
*Are you interested to cook?
There is, at least in American English, no discernible difference as far as meaning between the two phrases. Both phrases introduce something which you wish to learn. However, the form interested in knowing is a bit more common that interested to know, which does sound a bit more awkward (at least in my opinion).
Another phrase, if you're curious, would be interested in learning.
I am interested in learning whether or not dogs have legs.
Based on my intuition and the time I've spent trying to understand Jean Yate's The Ins and Outs of Prepositions I would suggest that 'in' gives a more detailed interest in learning, perhaps a specific detail or for a specific goal' whereas 'interested to know' is perhaps a general interest, but maybe nothing more.