Example #2 (the latest week) is a "timeless" reference. That's to say, it could in principle refer to the most recent of any contextually relevant set of weeks, whereas example #1 can only refer to the week immediately preceding "time of utterance" (or "time of writing", here).
There is more to think about, though. It's currently Friday afternoon July 19th. If I write...
A: I earned £1000 last week
B: I earned £1000 in the last week (or the past week, which is equivalent here)
...most people would probably understand A as meaning I earned that much during the week lasting from Mon 8th - Sun 14th (I don't want to get bogged down in whether weeks run Mon-Sun, Sun-Sat, or whatever). But B would normally be understood as referring to my earnings this week (being the most recent working week which is "more or less complete" by Friday afternoon).
But note that I could precede B above with some "non-current" temporal context, such as...
C: I've just finished my accounts for June. I earned £1000 in the last week
...which would mean I earned that much in the final week of June. Note that it wouldn't make sense to do this with version A above. Nor would it be possible to replace last with past in C, because the past week always means the same as last week (they're both always relative to today, now).
Superficially this may look confusing, since the last week in B is later than last week in A, but it's much earlier than that in example C. The way to look at it is first to recognise that article-less last week is the standard way to reference the week preceding the one during which an utterance is made (that week is definitely in the past, since we're now in a later current week).
If a native speaker chooses to include the article, he's "overriding" that standard sense to some degree. Usually this is because he wants to reference the current week which is now just ending.
The latest week simply means the most recent of the contextually relevant weeks. It wouldn't normally be used in contexts A and B above, where last is the natural choice (weeks considered relative to today), but it could be used in example C (where it would refer to the latest of the June weeks for which I did my accounts).