Is the simple future used in spoken Portuguese?
Yes, it certainly is. But this needs to be qualified.
The future simple has many values. Here are some, and how they could (or not) be replaced with ir (present) + infinitive, haver (present) de + infinitive or the present. (There is also the form ir (future simple) + infinitive, which is similar in meaning to the simple future.)
- Quando chegar a altura, tratarei do teu funeral. (distant future without modality; fully replaceable with the present trato; replacement with hei de tratar adds an odd enthusiasm to the sentence; replacement with vou tratar suggests a firmer intention)
- Amanhã viajarei para o Porto. (simple futurity; replaceable with vou viajar and, in this case, with the present viajo because the adverb "amanhã" eliminates any ambiguity)
- Chegará o dia em que isso deixa de funcionar. (tone of prophecy; most commonly replaced with há de chegar; also replaceable, to a lesser extent, with vai chegar, which maybe suggests the day would be very close)
- Nunca o direi. (promise/commitment; replaceable with hei de dizer and, less emphatically, with vou dizer)
- Pagarás pelo que fizeste. (threat; replaceable with vais pagar and, to the extent that it is in an undeterminate future, hás de pagar)
- Farás o que eu te digo. (deontic modality; replaceable with vais fazer with the right intonation)
- Não matarás. (one of the Ten Commandments; similar to above but non replaceable with vais matar)
- Por esta altura estará já em casa. (probability, episthemic modality; replaceable with há de estar)
- Não havia sinais de entrada forçada, pelo que o ladrão terá entrado pela porta das traseiras, que não estava trancada. (logical conclusion of which one is not 100% certain; replaceable with há de ter entrado, which would give it less certainty; with entrou would sound like a certain conclusion)
- O detido será o autor de um homícidio. (alleged fact; replaceable é provavelmente/alegadamente/presumivelmente)
- Compreenderás que não o fiz de propósito. ("I hope you understand..." but a little more suggestive or forceful; replaceable with Hás de compreender)
- Onde é que estaremos? (rethorical question, suggests a problem; not replaceable)
- Será que ele já chegou? (wondering; it is a slightly more common version of "Ele terá já chegado?"; not replaceable)
Of these, the sentences expressing doubt or possibility are the most likely to be used in spoken Portuguese.
The usage is also different between Brasil and Portugal. Generally speaking, the ir + infinitive is more grammaticalized there (there's no idea of immediate future or going somewhere) and the periphrasis haver de barely exists. Translating from here:
The form ir + infinitivo in Brazilian Portuguese is more grammaticalized and in a more advanced evolutive state than in Spanish and in European Portuguese. This fact can be observed not only in that it has turned into the form of choice to express futurity (distant or immediate), uprooting the synthetic form -rei in the oral language, but also because it's being threatened by other forms of expressing futurity like the periphrasis estar + gerúndio, which is unheard of in Portugal, to express immediate futurity and to express intentional modal content, given that these values are weakened in ir + infinitivo in the Brasilian Portuguese norm.
Then there are sociolinguistic factors. More educated people tend to use the simple future more. On TV, the typical football coach probably won't use the future simple (except in "será que" and other limited circumstances), but more educated people will.
The general tendency is for ir + infinitive when it can be used.
Are there any situations where the two might have slightly different meanings?
For someone who's not an advanced learner of Portuguese, I wouldn't worry too much about the differences between the two except for the situations where the future simple denotes probability/doubt, which the periphrasis with ir cannot.
Nevertheless, as hinted before, there are also other differences, but only in European Portuguese.
The periphrasis tends to express:
- A prospective aspect, including (but not limited to) ingressive (beginning of a new action) and inchoactive (beginning of a new state) aspects.
- A near future rather than a distant one.
- An intention rather than a quasi-expectaction.
A prospective aspect means the event time happens later than the reference time. Take @Jacinto's sentence:
Em Janeiro, já serei pai.
By January, he will already be a father; the future event started before the reference time (January). "Ir" is indeed possible, but less natural. You could also say:
Em Janeiro, já hei de ser pai.
Which is similar, but introduces more distance or detachement in the statement.
Compare also these two sentences:
(1) Amar-te-ei sempre.
(2) Vou amar-te sempre.
Here (2) is merely prospective (maybe you didn't love them before), and can mean that you will start loving them now. (1) is more compatible with a meaning where you've always loved them and will continue doing so.
Let's also compare these sentences:
(1) Eu farei o trabalho.
(2) Eu vou fazer o trabalho.
We can interpret the contrast in several ways:
- (1) is a mere statement of fact, where (2) signals intention of doing the work,
- (1) indicates you will do the work at some point, where in (2) you will be doing the work in a near future,
- (2) indicates that you will begin doing the work.
Note also that, with the right context, their differences are mostly flattened:
Amanhã farei o trabalho.
Amanhã faço o trabalho.
Amanhã vou fazer o trabalho.
(Amanhã hei de fazer o trabalho doesn't mean the same, it is more of a commitment or reminder to yourself.)
By the way, you've asked very broad questions, if you want to know more, you can look at the thesis I mentioned above.