Basic dictionary definition; asylee vs refugee. refugees apply at embassies, asylum seekers apply while they are already here.
Some official numbers from USCIS https://www.dhs.gov/immigration-statistics/refugees-asylees
2016 report https://www.dhs.gov/sites/default/files/publications/Refugees_Asylees_2016_0.pdf
Basically US accepts significantly higher number of refugees (85K) than granting asylums (12K). Top countries for 2016: refugees came from Congo, Burma, Syria, and Iraq around 10-15K each; whereas asylums granted to El Salvador, Guatemala, and China around 1300-1400 each.
Here are some numbers from currently pending asylum cases (March 2018) https://www.uscis.gov/sites/default/files/USCIS/Outreach/Notes from Previous Engagements/PED_AsylumOfficeWorkloadMarch2018.pdf
Currently over 300K asylum applications still pending. That's a lot of people to fit in the 12K green cards/year.