In most contexts, I am going to [verb] and I will [verb] are interchangeable. Sometimes the former may place more emphasis on the fact of your current intention/expectation, where the latter emphasises the future action.
There are some contexts where the difference is clear, and this may have some bearing on why OP thinks there's a planned/spontaneous distinction (usually there isn't). Say you're round a friend's house watching a football match on TV, and at half-time the beer runs out...
You: "I'll nip down the shop and get some more beer."
Friend: "It's raining - I'll find my umbrella for you."
If you reply "Don't bother - I'm going to use my car", the implication is you had already decided you were going to use the car before you first said you'd get the beer. But if you say "Don't bother - I will use my car", this implies you just made that decision in response to what your friend said.
@Peter Shor mentions another context ("It's going to/It will bite!") where native speakers often make a distinction. In that case, going to usually signifies immediate danger (it's just about to bite), but will can just mean it's bitten others before, and will/may bite you soon if you're not careful (effectively, the same as "Careful! It bites!" where present tense indicates "habitual" action).
Kosmonaut's "Let's say that tomorrow you will walk your dog" isn't really relevant to the current issue. It could just as well have been "...tomorrow you are going to walk..." or even just "...tomorrow you walk...". They all mean the same, and in that context there's no real reason to prefer one over another.
Best Answer
"I'm going to go to the store" = I will go to the store in the future; I am not leaving now. It would normally be followed by a time, e.g. "tomorrow".
"I'm going to the store" could be something I am currently doing, about to do, or will do in the future. It's not idiomatic at all for "currently", and it would normally be followed by a time: "now" or "tomorrow" for example.
Idiomatically, if I wanted to tell someone I was currently going to the store, I wouldn't use the verb "to go". So I might say "I'm en route to the store".
Idiomatically, if I wanted to tell someone I was about to leave for the store, I'd say "I'm off to the store."
All this is the highly subjective opinion of a native British English speaker (me). And, to be honest, if I'm talking idiomatic British English I wouldn't use the word "store", I'd talk about a "shop".