In response to a recent FoI request the DWP have stated:

..... we are reacting to an external change in mail sortation that could lead to elongated delays in the customer journey, which we want to avoid
at all costs.
The way to avoid this is to move to white envelopes later in the year.
From other sources they will still be window envelopes to show the address printed on the letter.

I believe they are still deciding what to print on the outside of the white envelopes so that you know it's from the DWP about your benefits, without telling anyone else that you are on benefits.
That's pretty impossible, which is why the brown ones don't have anything "DWP" printed on them other than the return address on the back.
But they can have some all-expenses-paid meetings to talk about it.

TBH it's just going to make them harder to tell from all the (other) junk mail that comes through the door.