The meanings of the two terms largely overlap. Frankly, when they are used together, I think it's pretty much a redundancy for emphasis. If you asked a salesman, "What are the most important features of your product?" versus "What are the most important features and characteristics of your product?", I would expect you would get the same answer either way.
We do this a lot in English. Sometimes a single word just seems too abrupt. So instead of saying, "When the letter arrived, Sally was happy", we might say "When the letter arrived, Sally was happy and joyful." Instead of just saying, "Bob is irresponsible", we'll say, "Bob is lazy and irreponsible." Etc. Sometimes the extra words arguably do add some shade of meaning, but often people add extra words that mean pretty much the same thing just for emphasis: instead of saying "she was very very happy", it sounds more literate to say "she was happy and joyful".
You wouldn't normally talk about the "features and characteristics" of a person. This is a phrase used to describe inanimate objects, usually some machine or gadget. If you asked, "What are Mary's features and characteristics?" it would sound quite strange, like you were describing here as a product that you were going to sell.
When talking about a person, "features" means his physical appearance. If someone asked, "What did you notice about Charly's features?" they would expect an answer like, "He has brown hair and a scar on his left wrist." Similarly if you ask about the "features" of a place, like, "What are the features of the Toutle River Valley?", they would expect you to describe physical (geographic) features, like "There's a hill on the north end, a deep gorge running most of the length," etc. But if you talk about the "features" of a product, you normally mean details about how it functions. If someone asked, "What are the main features of your new Whizbang 300 cell phone?", they wouldn't expect you to answer, "It's black and sort of rectangular", but more like, "It has a built-in GPS and a function to mask out background sounds when you call your wife from the bar."
If you talked about the "characteristics" of a person, that could mean anything about him, from "He has blue eyes" to "He is an excellent basketball player." I suppose you wouldn't use "characteristics" to describe something very temporary, like "He is sitting at his desk"; it would normally be used only to describe a fairly long-term attribute.
(Whew, that answer was longer than I intended.)