Simple future (futur simple) and future perfect (futur antérieur) both help describe actions situated in the future.

The future perfect is used for an action that precedes, or will be completed before, another action in the future simple. The future perfect expresses the fact that the action will be perfected (accomplished) by the time the other action takes place.

• Il parlera dès qu’il aura fini de manger :
  • 1 → Il finira de manger.
  • 2 → Il parlera.

One action follows the other, they don’t occur at the same time. First he’ll have his meal, then he’ll make his speech. He might even leave the table to make his speech.

• Il parlera dès qu’il finira de manger :

Both actions are simultaneous. When he makes his speech he’ll probably have reached the end of his meal but he might not have finished it yet. He might still be sitting at the table, eating and speaking at the same time (or at least alternately).

Obviously simultaneous actions are not always possible. It is not a question of tenses used but of vocabulary.

If you say:

• J’irai me coucher dès que j’aurai fini de faire la vaisselle.

It is obvious I won’t do the washing up in bed, saying J’irai me coucher dès que je finirai de faire la vaisselle although comprehensible would sound weird because you can’t finish doing the washing-up in bed.

Let’s take another example than the one in the question without the verb finir.

• Il parlera dès que son père sera arrivé.
• Il parlera dès que son père arrivera.

In those two sentences we have two consecutive actions:

• 1 → Son père arrivera.

• 2 → Il parlera.

• Il parlera dès que son père sera arrivé.
  → The use of the future perfect focuses on the action being completed, it might take more or less time for the father to settle and for the son to start speaking.

  • Il parlera dès que son père arrivera.

  → The use of the future simple informs us just about the arrival of the father. Whether he’s on the threshold or already fully inside is not important. The actions are then perceived as occurring closer together in time.