Running (or installing) Ubuntu usually means you put it permanently on your hard disk, either by itself or alongside Windows, etc. You use it every day, all your documents/settings are saved, just like Windows.
Trying Ubuntu usually means you start it from a Live-CD or a Live-USB. You can use it for the duration of that session, but your saved documents/changes/etc. are lost when you shut the computer down (although you can set up "persistence" to prevent that).
Running Ubuntu from a USB drive is possible, either using persistence or actually installing to the drive as if it were a hard drive. The downsides include:
Slow: Regular USB "stick/flash" drives are nowhere near as fast as hard drives, and it will show, possibly detracting from your experience.
Lower life:: The frequent writes/swapping done by an operating system may be injurious to the health of your flash drive. e.g. it may only last one year instead of a normal five.
Limited space: If you intend to use Ubuntu for a decent period of time and/or use it for multimedia (movies, music, etc.), you will eventually run out of space even if you get, say, a 32 GB flash drive. That doesn't happen with hard drives, which are typically much larger.