What's the difference between "People are all doing ..." and "People are together doing ..."

First of all, there is only one couple in your example sentence.

To be grammatically correct, the sentences would start with a different form:

That couple is eating a dish.

But while I could rewrite it to compare both and together with respect to the individuals, it's simpler to use a plural form.

I'll also assume that the word dish is not being taken literally, and that what's actually being eaten is a single plate of food.

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Those couples are both eating a dish.

This means that there are two dishes. One couple is eating one of the dishes, while the other couple is eating the other dish.

Another way of phrasing this would be to use each instead of both.

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Those couples are eating a dish together.

(Note that I have moved the location of together in order to make the sentence less awkward.)

This means that there is only one dish and it is being shared between the couples.

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For the sake of completeness, if you wanted to reprhase this to talk about a single couple, the two sentences could look like this:

The two members of that couple are both eating a dish.
The two members of that couple are eating a dish together.

The same analysis of the meaning would apply here.

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Note that the specific word dish makes a significant difference to the common meaning of the sentence.

The couple shared a meal.

Idiomatically, it's more common for this to mean that they ate at the same table together, each person eating their own dish.

Or:

The couple ordered the same food.

This would commonly be taken to mean that they ate at the same table together, each person eating their own dish, and that they had each ordered the same menu item.

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Part of the problem with your question relates to the words you chose, which may be leading you astray.

There are also at least two different senses of together—one means a combined effort or activity (which is the sense I used above), while the other means in close physical proximity.

You would not normally say that couple ate at a table together because, if they are a couple, they normally would eat at a table together.

But you could say those two couples ate at a table together.

The fact that the examples used in the question involve a couple and eating confuses the matter. The difference between both and together is obvious if one considers a different pair of examples, such as:

Jack and Jill are both doing their homework.

Jack and Jill are doing their homework together.

The second logically entails the first, but not v.v. The first, but not the second, is compatible with their working separately in their respective rooms on the items of homework that have nothing to do with each other. The second would most likely be used in a situation in which the two of them are collaborating on the same homework; at a minimum, it implies that they are in the same room, and that they are in some way supporting each other in their work.

The question also asks 'What is "both" for?' in the sentences of the first type. Both is semantically redundant: 'Jack and Jill are both doing their homework' is true if and only if 'Jack and Jill are doing their homework' is true. Both nevertheless does have a purpose: it is an emphasising device.