This answers it: https://www.reddit.com/r/askscience/comments/5gae8u/why_are_...

"Ice crystals are like a six-sided prism. This prism grows as more ice molecules stick to its faces. It turns out that under conditions found in common snowstorms, some facets in XY plane tend to grow much faster than the facets along the main axis of the crystal. As a result, snowflakes usually end up looking like flat pancakes with many finger-like branches"

The edges grow much faster because "molecularly flat regions... have fewer dangling chemical bonds and are thus less favorable attachment sites [for condensing molecules]." Also, snowflakes can take on 3D column shapes depending on humidity and temperature. See http://www.its.caltech.edu/~atomic/publist/AmSci2007.pdf

Right, but from that we might also expect very long thin hexagonal crystals. Maybe that is what rime is?

There are some shots of those near the tail end of the article. Both pure cylinders (https://c6.staticflickr.com/9/8240/8661684397_3e2953b2ca_b.j...) and capped columns (https://c8.staticflickr.com/6/5613/29417525663_448be7dac8_o....).

Shooting from the hip here, but another hexagonal crystal, graphite, is strong laterally but not between its lattices.