Routines and modes are often used together but they are two different things.
A routine is a set of instructions that run one time and then are completed. You can schedule them to run over and over on different days or however you want to control it, but each time the routine runs just once. So if you see the notification in your mobile app that says the routine ran, it ran and it finished. (Hopefully, sometimes there's a glitch and not everything runs but it's supposed to.)
The mode is a global state variable for your location. It can be set to any value you choose. For example, at my house we have a value of "night" and we have another value for "asleep." When the mode is night, triggering a motion sensor in the bedroom will turn on the overhead light. When the mode is asleep, triggering that same motion sensor does not turn on the overhead light – – instead, it turns on a soft night light on the wall.
Typically the mode is changed by a routine, although it can also be changed by a smart app.
You can see the mode value that is currently set by opening the mobile app, clicking on the three horizontal lines in the upper right, and the mode will show there. I can't remember if it also shows anywhere else. (Since the last iOS SmartThings update, I have been unable to access the mobile app since it no longer allows for voiceover navigation.)
So for your first question, how do you know what routine ran most recently, you open the mobile app and then look at your notifications.
For your second question, how do you know what mode your house is currently in, you open the mobile app, click on the menu icon in the upper right, and you will see the current mode value.