When it comes to understanding future forms, context is everything.
This context sounds like the beginning of a presentation, lecture or talk. You could say either 'Today, I'm going to show you..' or 'Today, I'll show you...'. Neither one is spontaneous, as the speaker has clearly prepared in advance what he or she is going to show to the audience. In this case, there is no difference in meaning.
In other contexts, 'I'll' can indicate spontaneity. For example:
"One decaf coffee, please"
"Sorry, we don't have any decaf."
"Oh. Right. Well...um.. I'll have a peppermint tea instead".
In the peppermint tea context, you couldn't use 'going to', because the speaker has only just made this decision at the moment of speaking. In other contexts, such as your presentation above, you could use either.
By the way, 'gonna' is not a future tense. You will not find this 'tense' in any grammar book. 'Gonna' is merely a transcription of the sounds which we hear when we say 'going to' in a relaxed way. The future construction is 'am/is/are going to' - 'gonna' is just what it sounds like.