From a certain point of view, of course, the two are essentially synonymous. The two words convey the same sense of what is desired or required. But from the legal standpoint, as made explicit in United States immigration law, the two are not synonymous. If a person is qualified before entry into the US as a refugee, that person cannot be an asylum seeker in the legal sense even if the words are presumed to mean the same thing. An asylum seeker is, by definition, a person who has not qualified for admission to the US through its regular immigration channels. Refugee status is based, in part, on membership in some pre-defined class. Asylum is individual from the start. If, for example, the government decrees that nobody from Country X may be admitted to the US, then no refugee can be admitted to the US. But asylum seekers, if they make the journey, can and must be given consideration as potential refugees, no matter where they come from.
Language aside, according to US immigration law, it is legally not possible for a person called a "refugee" to be the same as a person called an "asylum seeker." And an "asylum seeker" is not a refugee unless due process occurs and refugee status is granted. This distinction is important because it gives (or should give, in theory) a special status to asylum seekers, in that their urgent need for safety trumps the process of immigration law, diplomacy, and other factors that are weighed in the decision to grant refugee status, including the very possibility of documented admission.
You seem at times to be adamant in stating that US policy trumps international principles and declarations. So it seems a bit odd that suddenly the reverse should be true. You can look up the US immigration policy.