Fold the paper differently to change the symmetry. You can have
* 4-fold symmetry (fold in half and then half again),
* 6-fold symmetry (fold in half, then into thirds),
* 8-fold symmetry (fold in half, then into quarters),
* 12-fold symmetry (fold in half, then into thirds, and then fold in half again). At this level, thin paper is a must.
Keep in mind that real snowflakes have a six-fold symmetry and sometimes a 12-fold symmetry. If you fold the paper into four or eight sections, it shouldn't be called a snowflake anymore.
Make lots of small cuts instead of a few large cuts. More cuts will give a more lacier look.
Make long cuts that go deep towards the other side of the paper. This will give you an elegant design instead of the chunky kindergartener look. Be careful though, too much cutting will cause the entire snowflake to fall apart.

Bekah Gjerde's Flicker photos
Kanako Yaguchi's web site
David C. Stredulinsky's web site
GW Hart does modular kirigami

