Learn English – Difference between “will do” and “be doing”

It's basically identical, yes. The phrase suggests that the thing being done is slightly difficult or laborious in some way, or perhaps that is has previously been delayed for some reason, but it's very subtle. This is likely to be spoken English, not written.

BE + going to - Lindsay is going to fly to New York next week.

Forms with BE + going to possibly originated in such utterances as:

1. We are going to meet Andrea at the cinema,

uttered when we were literally going, i.e. on the way, to the meeting. At the moment of speaking there was present evidence of the future meeting. This use has become extended to embrace any action for which there is present evidence – things do not have to be literally moving. Consider now these two utterances:

2. Look at those black clouds. It's going to rain.
3. Luke is going to see Bob Dylan in concert next year

In [2] the present evidence is clear – the black clouds. In [3], the present evidence may be the tickets for the concert that the speaker has seen on Luke’s desk, or it may simply be the knowledge in the speaker's mind that s/he has somehow acquired.

Modal (will) - Lindsay will fly to New York next week.

Will is a modal and, like the other modals, has two core meanings. The two core meanings for most modal are:

(a) the 'extrinsic' meaning, referring to the probability of the event/state (b) the 'intrinsic' meaning, reflecting such concepts as: ability, necessity, obligation, necessity, permission, possibility, volition, etc.

The extrinsic meaning of will is exemplified in:

4. Emma left three hours ago, so she will be in Manchester by now.
5. There will be hotels on the moon within the next 50 years.
6. The afternoon will be bright and sunny, though there may be rain in the north.

In all three examples, the speaker suggests 100% probability, i.e. absolute certainty. (MAY would imply possibility, MUST logical certainty, to take examples of two other modals). Note that while certainty in [5] and [6] is about the future, in [4] it is about the present. It is the absolute certainty, in the minds of speaker/writer and listener/reader, that can give the impression that forms using ‘the will future’ are some way of presenting ‘the future as fact’. Some writers therefore call this form ‘the Future Simple’. Weather forecasters, writers of business/scientific reports, deliverers of presentations, etc, frequently use will, and learners who encounter English more through reading native writers than hearing native speakers informally may assume that it is a 'neutral' or 'formal' future. In fact the particular native writer or speaker is simply opting to stress certainty rather than arrangement, plan or present evidence.

The intrinsic meaning of will is exemplified in:

7. I'll carry your bag for you.
8. Will you drive me to the airport, please?
9. Jed will leave his mobile switched on in meetings. It's so annoying when it rings.

These examples show what we might loosely call volition, the willingness or determination of the subject of the modal to carry out the action. Note that [9] is not about the future, and in [7] and [8] the futurity is incidental. It is context rather than words which gives the meaning.