So on a large scale many people misconstrue what a Sociopath and a Psychopath are. This misconception of them is often derived from the fact that many psychopaths exhibit sociopathic traits however not all sociopaths are psychopaths.
Lets start with defining the terms
Sociopath - a person with a personality disorder manifesting itself in extreme antisocial attitudes and behavior and a lack of conscience
Psychopath - a person suffering from chronic mental disorder with abnormal or violent social behavior
Now the part that it seems gets confusing is the last part that describes a sociopath "lack of conscience". This isn't to say that a Sociopath will walk out of their house and just kill someone this just means that their moral compass doesn't necessarily point North. In fact, many sociopaths know the difference between right and wrong and a majority of them will adhere to it as it is socially acceptable.
On the flip side, sociopaths are more rational than emotional. That means that if the choice is between a dog that "might" die and a child that "will" die they may choose the dog over the child because rationally it makes better sense to give the possibility over the sure thing. Additionally, sociopaths are more likely to feel empathetic toward animals than toward humans (re: antisocial behavior).
Now, many people mistake the lack of empathy for no feelings whatsoever. Sociopaths have feelings, just like the rest of us, they just don't interpret their feelings in the ways that someone that is not a sociopath interprets that data. For the most part a sociopath will feel little to no empathy toward humans or their empathy will be misdirected. I have a quote from a diagnosed Sociopathic friend (at least I call them a friend) that told me the following:
While they were diagnosing me they showed me pictures of many things and one of the things they showed me was a picture of a Nazi war camp. The picture had prisoners about to be executed. When they asked me about how it made me feel I told them "I feel bad for the soldiers". They asked me why. I told them "Because they were made to do all of those horrible things."
This person loves dogs and has a really connection with them where they don't have this same connection with other people.
Many sociopaths are relatively well adjusted. They take social cues decently well and are able to blend in. Many sociopaths that you see in media are the exact definition of "antisocial" and often the reason that it's blurred into the next subject.
Ah... psychopaths! The overused term for the wrong type of character. Many people believe that the above description is actually a psychopath however psychopaths are their own demon by themselves. First and foremost you do not have to be a sociopath to be a psychopath however many serial killers are both. Additionally, and most importantly, not all sociopaths are psychopaths. A good way to remember the difference between the two is that Psychopaths want to hurt people, sociopaths do not.
Psychopaths are aggressive beings, this is unlike the sociopath who is anti-social and for the most part uncaring. These are often the people that are portrayed in movies to kill small animals while they are young. Psychopaths are extremely suave, intelligent and blend in very well with others. It is rumored that a lot of psychopaths are as such because they are so smart but that is a speculation.
Psychopaths can have other mental disorders that confuse the person and make them more radical than others (in opposition to the suave and intelligent statement I said early). This can manifest in many ways and many other mental disorders, that are also confused with psychopathy, including schizophrenia manic disorder and more.
Overall, psychopaths are misrepresented because they are confused with many other mental disorders that are often blurred in media and other avenues. Psychopaths are very specific and should not be confused with sociopaths.