In the likelihood that someone who knows will swing by: why are some UK government departments called Departments eg Department of Work and Pensions, and others Ministries eg Ministry of Defence, Ministry of Justice. It can’t be a historical thing, I think, because the MofJ is a recent creation.
I think it comes down to whether the senior politician is a Minister or a Secretary of State. Ministers sit in Cabinet vs Secretaries who didn’t, or at least this distinction mattered at some point in the past…now I believe it’s just an interchangeable term and little more.
Cabinet-level appointments are (mostly) both Ministers and Secretaries of State though. 
Perhaps someone should have written all this stuff down at some point in the last 800 years…
Well, quite. I think back in simpler times officials tended to be one or the other, but now they wear multiple hats…
I think it was just a matter of Prime Ministerial whim over the last five or six decades as to whether particular policy areas (and/or individual political personalities) justified their being made to sound as grand as the established “great offices”. I don’t think it’s any older than Harold Wilson and Ted Heath.
It’s really only since the First World War that central government and its organisations have become so big and diverse as to make all this even possible. Time was, some areas weren’t even ministries or departments - there was, for example, the Board of Trade and the Board of Health, and the minister appointed to be in charge was “President of the Board”, or there was the Postmaster-General, who ended up de facto responsible for policy on broadcasting. Plus there are or were a few old mediaeval titles that were used for assorted multitasking (usually presentational/politicking, rather than departmental management) jobs, like Lord Privy Seal and Chancellor of the Duchy of Lancaster (though they may also have had residual ceremonial functions to do every now and again).
We can even have a “Minister without portfolio” who sits in the cabinet but does not have a department to run.
No, there is a highly technical distinction between the two. Secretaries of State can do one very specific thing that most other ministers cannot. That is bound up with the origins of the position which so far have only been briefly touched upon in this thread.
Originally the monarch just had a secretary to help them with their paperwork. During the sixteenth and seventeenth centuries this evolved into the two Secretaries of State, at least one of whom would be a major figure, possibly even the monarch’s de facto chief minister. Both Secretaries handled both the foreign and the domestic paperwork (although confusingly they often divided this geographically between the north and south of England and the north and south of Europe). In 1782 this was altered so that one handled the foreign correspondence, the Foreign Secretary, while the other handled the domestic correspondence, the Home Secretary.
A crude simplification would be that over the next two centuries these duties were further divided, with more specialised Secretaries of State being appointed to handle specific policy areas, usually ones formerly dealt with by the Home Secretary. But in reality the process was much, much more complicated than that. It would be more accurate to say that other domestic departments were created at various times with various titles being used for the government ministers heading them and that most of those ministers with Cabinet rank have since been given the title of Secretary of State just for the sake of consistency.
But there is one advantage in being a Secretary of State. That dates back to the time when there was just two of them. Certain types of government orders need to be signed by the monarch. Needless to say that is now just a formality. But that signature (the ‘sign manual’) has to be witnessed by one of Secretaries of State or by one of the Lords of the Treasury. This was originally intended not so much as a way of the ministers controlling what the monarch was signing as of coordinating what was being signed - you didn’t want the King to be signing contradictory orders. However, in modern times it has been found useful for each of the major executive departments to be headed by someone, a Secretary of State, who can present such documents to the Queen for her to sign. They might not have to do this very often, possibly not even ever, but it is a convenient right for them to have when the occasion arises. I very much doubt most of them have the slightest idea about this.
As it happens, it is still often the Home Secretary who does have to do this. Because the Home Secretary originally handled all the domestic paperwork, they have sometimes been left dealing with some of the formal stuff that is too minor to be of any real interest to other departments. Also, if such stuff was actually important, less cumbersome means would been found by which to enact them. In that respect, there is a certain similarity with some of the more arcane duties of the US Secretary of State.