In four short lines, the speaker tells someone they love that their fingerprints are more unique than the speaker's own. It's an odd compliment, since fingerprints are supposed to be one of a kind for everyone, but the speaker feels that the loved one's are somehow more special. The snowflake image makes them delicate, rare and fleeting, something that melts on the tongue. Yet the speaker adds that they're "somehow sweet," which turns a cold image into a tender one.
The last line carries the real feeling. The speaker asks to have those fingerprints "burned" into them, so they can be unique too. The word "burn" is intense and a little painful, which suggests a love that is hungry and wants to leave a permanent mark. The speaker seems to feel ordinary and hopes to be changed by being touched. The mood is intimate, yearning and slightly vulnerable, with a quiet edge of devotion.