What is the difference between "sign" and "symbol"?

There is a broad overlap between sign and symbol; it's a nuanced distinction.

In your examples, sign should be read as a synonym for evidence. More broadly, there is an intrinsic link between the sign and what it signifies. On the other hand, a symbol is merely a declaration that has an imposed relationship with what it declares.

For example, one might say that a smile is a sign of contentment if it is accepted that contented people tend to smile. One could also say that a smile is a symbol of contentment, but that would be a relationship between a smile and the concept of contentment (an imposed relationship - concepts don't smile), not between a smile and a contented person (an intrinsic relationship).

If, in a hypothetical culture, bananas were placed in front of contented people, one might say that a banana is a symbol of contentment, but it would be weird to say that a banana is a sign of contentment because there is no intrinsic link between contentment and bananas (contentment doesn't produce bananas).

So in your first example, I'd call meat with hair a symbol unless you read a mystical component into the context. The bananas in your second example are definitely symbols - they represent a wish and are not a by-product (or herald) of happiness.

Personally I'd start from Peircian Semiotics to pick apart the nuances and make sure that I meant what I meant to mean.

Anyway, I'd stand sign in for Peirce's index. A sign points to some specific instance of something [or some thing]. A symbol stands in for the idea [and perhaps idea could be taken in the Platonic sense to some extent]. The world of symbols is complex: characters on screen stand in for sounds -> sounds stand in for words -> words stand in for things in the world -> things in the world may be symbols. Normally signs don't point around in circles.