"character", unlike characteristic, is never used to describe a single feature. You can say things like "This man/woman/place has character, and the word here refers to all the features that makes this man, woman or place interesting and different. When you're refering to a single trait or feature that someone or something has use "characteristic".
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.