It's been a while since I've seen Season 3, and maybe that'll make me a better reviewer (more likely a worse one). This season is where the Queen appears to kick into high gear, and settle into her personality that we know today.
The Queen is older in this season, and in middle age, she questions whether her country still needs her given its rapid changes and modernization. Given this, she throws herself into her work. This leaves her family rather high and dry, which is something different from the prior innocence we've seen in Seasons 1 and 2. Charles is started to be ignored (and this continues for the rest of his life), Margaret becomes rather embittered because her marriage (which she dove into in the shallow end for and hit her head because the one she really wanted couldn't marry her) is falling apart and also becoming middle aged, and Lord Mountbatten tries to overthrow the government because he's old and stuff.
You can kind of start to see the outlines of the modern Royal Family as we see them today in this season, and the modern struggles that still befall them to this day. I think it's well worth a watch.