I am not sure the two are really different things.
Is not the ability to write just the delivery tool for telling the story?
I think the ability to write just allows you to deliver the story better.
I guess this depends on how you define ability to write.
To me, in this context, being able to write means that I can express myself with clarity in writing. I can get my point across to the reader.
Defining storytelling is trickier, but lets say storytelling is describing a series of events. Based on that, we can define good storytelling as describing a series of event in such a way that the reader will keep reading until the end.
The above is how I see it.
Another way of viewing it might be that you define writing a story as the act of designing and creating the reading experience. Seen this way, writing combines the ability to write and the ability to tell a story into one.
I wouldn't say that this is wrong, but it IS different from how I view it. I'd like to think that I am for the most part pretty good at the technical aspects of writing - the wordcrafting. I tend to get my point across, usually.
I'm a lot less certain about my storytelling. Are my characters as interesting to my readers as they are to me? Is the world as fascinating, the action as exciting, the pain as real?
Can my wordcrafting make my characters as interesting as I feel they are on its own? I don't think so.
I think that in order to do that, I have to have an understanding of what makes them interesting and why so that I can choose to write about that.
Maybe storytelling is what you write, and writing is how you write it?