Firstly,  the difference between those two is subtle enough not to worry students of intermediate level and below about it!

If we are going to teach the difference, then it's best  if we look at all four ways to talk about future, go through the examples in the text, and then create examples and practice them in their appropriate contexts.
For example: "Matthew's going into the city to a club tonight. He's going to hear his favorite band, and he's probably going to dance a lot."  Notice how the last two verbs use either  "will" or the " 'be' going to", whereas the first uses the present continuous for future.
We can use simple present to express the future, especially if we are looking at a datebook or itinerary. Then, we use the present continuous for the following statesments.
A present form expressing the future needs a specific time implied or stated, as in "Where are you going after class?" or "I'm going to the hairdresser's tomorrow morning." or "I'm seeing my cousin this week." or "Tonight we begin early."  The time cues are necessary to distinguish the intention of the future from that of present.
In the case of "am / are / is + going to," you do not need to give a specific time, as the focus can be on your intention, rather than on the time it will occur.   Of course, the same thing can be said for "will," but notice the extra immediacy expressed here:  "I'm going to take the cat to the vet." A time expression is appropriate if you have set a time for the action. "I'm going to take the cat to the vet right away." But look at this: "I'm taking the cat to the vet (now / right away.)
Good luck!