Okay, as a very personal take, absolutely unscientific and based on vague recollections and probably unjustified prejudices…
Wargames Illustrated is very visual and prioritizes the best looking games. Often themed, although theme may well be a bit wider and more generic than in WSS. Some good games and scenarios in there, although the very unscientific impression I had when last subscribing to this (maybe 3 years ago? I forget) was that it used the formula first established in White Dwarf where scenarios and in-depth battle reports would be linked. But this is a vague impression left with me, I could be wrong about then and it may not reflect current practice. Some good painting articles but pitched at the quality 28mm level. Both this and WSS seem to go bigger on the new and shiny games.
Miniature Wargames is the most varied in a given issue, and perhaps even taken as a whole. Good opinion pieces sometimes. The editor of Battlegames edited this one for a while and it retains some of that character (more below); probably features more Fantasy and SF overall than the others.
Wargames, Soldiers and Strategy is the most focused on a given theme, so a mixture of history, scenario, terrain building and painting (often to a very high "showcase" standard) and reviews built around that. Obviously that leaves a little less space for variety of coverage in a given issue. Regular and irregular columns are probably the best IMHO. Good comprehensive reviews, but perhaps marginally less challenging than Miniature Wargames/Battlegames' reviews are/were.
Battlegames very much prioritized the "game". So lots of versatile scenarios, terrain building and painting focused on nice but doable; quite a few articles on managing gaming projects, doing better gaming, lots of rulesets published (a couple of which I have actively used). Very comprehensive reviews section. Good opinion pieces, but not as good as those in WSS.
And such differences as there are are pretty marginal. Many of the same writers send (different) articles to different magazines. In the grand scheme of things, they are all pretty good. I think that there have been problems with all of the above magazines in the past, but these have all been sorted out a while ago.