In many ways, asylum seekers and refugees are similar. Both are facing some level of persecution or danger in their home country. This is often based on things like their ethnic background, their religion, their political stance or something of this nature. They want to leave their home country and come to the United States because they believe they will be safer from persecution if they are here.
That said, the way that they go about entering the country is much different. Seeking asylum is similar to being a refugee and accomplishes a similar goal, but there are different categories of people who are going to qualify, depending on where they are when they decide to take the next step.
Are you already in the country?
The biggest question is whether or not you are already in the United States, perhaps illegally, because you were simply fleeing persecution and crossed the border as quickly as you could. You could also be at a port of entry, trying to enter legally and hoping you can do so as soon as possible.
In these cases, your best option is to seek asylum. If granted, the United States will then let you stay in the country, even though your initial entry may not have been done correctly.
If you are outside of the United States, then it may make more sense to use refugee status. Refugees are vetted in advance and then given permission to enter the United States once their status has been granted. This is the main way in which it stands apart from asylum seeking, where you are already in the U.S. when you get the process started.
Your legal options
In both cases, the legal system can be complex. Make sure you know exactly what steps to take. It can help to work with an experienced law firm.