All green cards are the same, no mater how they are obtained.
Getting a GC thru asylum has a good thing, which is that it is backdated 1 year that counts towards citizenship.
There is another thing that most people are not too crazy about, and it is a code that shows on the card, (AS6, AS7, or AS8), and basically it means that the Permanent Resident Status was reached through Political Asylum, and that might raise some questions if you happen to travel to your COP. Other than that, all permanent residents share the same status.