Hi Gilberts,

Both terms do express the sense of "completion" but enfin can take the sense of "at long last" which is more nuanced than finalement which gives a sense of "finally" / "eventually".

The total text of the lesson expresses a number of activities over a period of hours after which the presents are opened   --- "at long last"  --> enfin.

This is how I see it  --  hope it helps.

Bonne journée

Jim

Just to complement Jim's answer:

Enfin and finalement can both be translated as "finally" in English, but they are not exact synonyms in French.

Finalement implies a kind of turn of events that was in some way unexpected, whereas enfin doesn't. Here's an example:

Je suis enchanté de vous rencontrer enfin. -- I'm pleased to finally meet you. This says that it's been a long time that you wanted to meet but now, in the end, you did meet.

Je suis enchanté de vous rencontrer finalement. -- I'm finally pleased to meet you. Here, you're saying that you actually hadn't wanted to meet the person but you're happy now that you've met. Note that in English the difference is conveyed by the placement of the word "finally".