Do they mean the exact same thing? Any nuances?
To borrow the Macmillan Dictionary's example:
Londoners came out on the streets in their thousands.
may well be phrased as
Londoners came out on the streets in the thousands.
in their thousands can mean a grouping of people, implying that they all came out for the same purpose. It can also mean a grouping as compared to other groups, i.e.,
The Londoners came out in their thousands, the Berliners in their millions, but the aliens came in their billions, and overwhelmed them.
in the thousands is specifically specifying quantity. Perhaps there is a cause for them to be numbered together, but their following actions may be different. E.g.,
The Londoners came out in the thousands. Some came out to escape the heat of their top-floor flats, while others came to observe narrowly the tawdry parade going before them. The children, of course, came out for the hard candies thrown out the sides of the colorful carriages.
It's written in their thousands because there are thousands of them.
In the thousands describes the number, not the people. You could say The Londoners who came out numbered in the thousands, meaning more-or-less at least a couple of thousands and (probably) less than a million. An awkward one; how do you describe one thousand, or a million?
A valid (in this context) alternative is to say Londoners by the thousand… although the literal meaning of this is that they appeared in groups of a thousand.
The danger of the former is that "in their XXs" can be taken as an indication of age, though we'd expect this to be more directly attached to the Londoners. Also by saying thousands (as opposed to e.g. 50s) it is fairly safe that age isn't really meant, but could cause potential confusion otherwise.
Agree with Tyaoshauo that "their" does add solidarity.