I don't think there is a way to concisely reply to this question (other than uninsightfully saying, ‘yes, there is more to the story’) so I will try to lay it out piece by piece. In French, there are three future tense:
- The simple future (le future simple),
- The near future (le future proche), and
- The future perfect (le future antérieur)
The third tense is out of out scope so let's just focus on the first two. Most simply put, the difference between the near and simple future is that the former happens in the immediate future, while the simple future happens further in the future.
LE FUTURE PROCHE (the near future)
As you pointed out, the near future is the English equivalent of ‘going to + infinitive’. This tense, as the name suggests, is used to express actions that are about to take place, or those actions which already have fixed intentions behind them and will take place.
- L’avion va atterrir dans 2 minutes. (The plane is going to land in 2 minutes.) - Most typical use, the action of the plane landing is about to take place.
- Je vais te téléphoner à midi. (I’m going to call you at noon.) - This action may not necessarily take place in the immediate future (i.e. the speaker might be saying this at 8 a.m.) but there is a definite intention behind it.
LE FUTURE SIMPLE (the simple future)
Again, as you pointed out, this is the English equivalent of ‘will’ future tense. This tense, at its basic level, is used to talk about future plans and predictions, and these events tend to be further in the future those for which we use the near future.
- J’irai à la banque demain. (I’ll go to the bank tomorrow.) - Indication of a plan to go to the bank, but note that if you use the near future (I am going to the bank tomorrow), then the level of intent or certitude increases.
- Je vendrai la maison l'année prochaine. (I will sell the house next year.) - Indication of a plan again and further in the future with limited certainty. Once again, if we use the near future for this example (I am going to sell the house next year.), then the speaker is determined that they will definitely sell the house next year.
Polite orders and requests. The simple future is used to soften commands or requests and often used in formal situations.
- Vous reviendrez demain, s’il vous plaît. (Please come back tomorrow.)
This is similar to our conditional use (i.e. ‘would you please come back tomorrow’) in English. This relates to your above example, because out of context (although it's hard to take this sentence any other way), Il n'y a plus de lait. Tant pis, je boirai mon café noir! sounds more like a request. It sounds more like, ‘there’s no milk? Oh well, I'll take my coffee black.'
With that in mind, if we compare it to the near future version, ‘I'm going to drink my coffee black’, then the impression I get is that the speaker is determined to have his coffee despite the lack of milk.
Si clauses. The simple future is often found in si clauses when talking about what will happen if a certain condition is met. For example:
- Je t’appellerai si j’ai des nouvelles. (I'll call you if I have any news.) - Here, we cannot use the near future because there is no certainty that the speaker will have any news.
- Tu réussiras à l’examen si tu étudies. (You'll pass the test if you study.) - Again, we cannot use the near future because the study is a hypothetical condition.
Future conjunctions. There are a few commonly used future conjunctions for which in English we would use the present, but in French we must use the simple future, NOT the near future. Here is a brief list followed by a couple of examples:
- après que (after)
- aussitôt que (as soon as)
- dès que (as soon as)
- lorsque (when)
- quand (when)
- une fois que (once)
- Dès que vous serez au lit, je vous raconterai une histoire. (As soon as you’re in bed, I’ll tell you a bedtime story.) - Note how in English the first clause is in the present tense, while only the second is in the future. In French, both must be in the future.
There are a few other instances where the simple future is used such as journalistic writing and other formal situations, but they seem irrelevant at this time. The simple future (will) has a wider range of uses than the near future, but the key differences between the two are:
- Timeline: aller + infinitive = near/immediate future whereas simple future = further in the future.
- Certainty/intention: aller + infinitive = a definite intention where simple future = a plan which may or may not come to fruition.
I feel like I have gone on a very long-winded explanation, but I hope it answers your question. In short, yes there are more uses and I hope they are clear.