First, let's start with what both Inspiration and feeling inspired means to me. Inspiration is something specific that you can point to say, “that was a direct influence on my project”. Generally inspiration comes before a project - For example, you may find several images to use as inspiration to help get aligned before a project starts.
Feeling “Inspired”, on the other hand, has more to do with the act of designing and is less easily defined as a specific thing you can point to. For example, you could say, “I was inspired by the clean lines of mid-century modern when laying out your deck design”. From the outside perspective, it may be unclear how, exactly, you took mid-century modern principles and applied them to a deck build. However, those were kernels which inspired your design direction.
Feeling inspired does not have to be as specific as a piece of “inspiration”.
You can let your mind wander, “go from the gut”. You don’t have to copy, you don't have to be exact. You do not need to memorize how the light came through that window you liked. You don’t need remember the exact size of the cool deck you saw last spring.
Why? Feeling inspired is about how your remember it - how it has sat with you and mixed with other ideas and experiences. You remember how it felt, and you will have the best parts of that experience at your disposal- you have your experience to use as a tool to create something new. Using a food analogy; being inspired is not a recipe or a specific dish, it’s the memory of the experience when you first had that great recipe.
So how to do get that feeling of being inspired? By experiencing things and noting how it makes you feel. How do you use it? Recreate that feeling... and that other feeling, and that one feeling from three years ago… Draw, Imagine, Design those feelings you experienced.
Bottom line, and to answer the question my post title poses - what’s the difference? It’s really a matter of how you want to use inspiration as tool or if you want to feel something before you start the act of design. Inspiration is great to get a start on a project or help sell your early ideas to a client or even your partner if you are working in a DIY method. Feeling inspired, however, has that artistic flair to it - the “get in the flow” type feeling where you use experiences of space, art, taste, feel, smell, etc. to make something new. And the mixing of inspiration over time becomes a way to be inspired.