Some of the four weather conditions that the UK has been through today! Grabbed from my Start screen during the last 12 hours:
The animations are, as I say, subtle. And, in the case of cloudy weather, non existent! But the sun's rays rotate slightly, the sunny spells fade in and rotate, while the rain err... falls! All very tasteful.
These weather updates are a trifle redundant, of course, since you can just look out of a window. But if it's currently sunny and the forecast changes to rain then I guess you'd better start looking for a shelter. More usefully, if you do tap through to Cortana's full pane then there's the option to tap through again for full hourly and daily forecasts.
All this also depends on you being interested in weather updates. If you're not, then go into Cortana and then tap on the 'hamburger' menu, top right. Under 'interests', you'll find a section on 'weather near me' - just toggle this off and you'll soon stop seeing the live tile and Cortana displays.
PS. All this is from me in the UK - exact Cortana behaviour may vary, depending on where you are in the world....