- How do people perceive the difference between these actions?
- How do people perceive the difference between a like/don't like system and a upvote/downvote system?
- When should you use which?
A major distinction is that you can usually see who 'likes' what, where-as upvotes are anonymous.
This is why the 'like' is mostly a social interaction. It's there to communicate to your peers that you approve of something or to show appreciation – sometimes it's not even the content that gets liked, but the act of posting by that particular person itself.
Similar to a real-life smile or a "Yeah, me too!" it's a low-effort tool that helps people bond while promoting opinions. Often used in networks where people can form inter-personal relationships (friend/follow) that usually don't go above 500 per user.
The upvote is often a lot more functional, it's there to evaluate content quality and importance through the power of democracy. Making them anonymous helps this cause, as there is far less social factors that one has to consider when pressing the button – very much like political voting.
You see the upvote mostly in networks where forming relationships is very secondary and it's the raw content that counts.
From the user point of view
The word chosen doesn't create much difference in user behavior. Such actions are motivated on an emotional level, and while users are very good at recognizing the valence of their emotion (is it positive or negative), they are not very distinguishing in what exactly they feel about an object. For example, if you have decided that you want an anonymous mechanism of endorsing, it won't matter to the user if you tell him/her to place a like, upvote, or five stars at the endorsed product; there are practically no circumstances under which the user would do one but not the other, as long as there are no other differences in the mechanism. Indeed, StackOverflow users have some difficulty distinguishing between the purpose of the voting, flagging and closing system and will frequently take the "wrong" action for a post they don't care much for, such as flagging an answer which has no technical problems, but whose content appears wrong.
But in case that you have different mechanisms, users will make a distinction. There will be cases where they would make endorsement under one system but not under another. Lukzen mentioned one large distinction between SO voting and Facebook liking: Anonymity. I guess that there must be more, but I don't have enough expertise in the area. If you are modelling your mechanism to be similar to an existing, widely used site, it is probably good to keep some degree of wording similarity to prevent confusion (but pay attention to not run into trademark problems!)
From the web site owner point of view
There are also many differences which users don't care for, or even hate if they are aware of them. For example, the Facebook Like button is used to track the surfing behavior of users on sites which have nothing to do with Facebook except of embedding the Like button. This is also a difference of the mechanism, not of the wording.
Generally speaking, a 'like' means that you like the post of subject matter, but don't necessarily recommend it. An 'upvote' is generally used when there are a large number of replies, questions, responses, or items that need to be ordered. The upvote means that the reader thinks that that piece of information is important and deserves to be seen first.