Is there a way to tell how the ice was formed? In other words, if you made a core through the arctic ice cap, could you tell how much of the core was from snow versus frozen sea water? What's the difference?

The Arctic Ocean loses 17 to 18 thousand cubic kilometers of ice every summer, only to regain most of that melted ice during the long Arctic winter when the ocean freezes at the surface. That corresponds to a layer of ice well over a meter thick that melts and reforms every year.

The Arctic is a desert, and an extreme desert during the Arctic winter. At 40 below zero (Fahrenheit or Celsius), it's simply too cold to snow. The amount of precipitation that falls on the Arctic Basin over the six to seven months or so when the ocean freezes is about a tenth of a meter. Snowfall cannot account for anywhere close to the meter+ thick layer of ice that forms during the Arctic winter. The tiny amount of snow that falls during the Arctic winter can't accumulate the way it does over ice caps.

Compare with Antarctica. Antarctica, like the Arctic, is a desert. Unlike the Arctic, Antarctica is a continent. The little snow that does fall accumulates over the millennia, and longer. The Antarctic ice caps have existed for millions of years. Now look at the Arctic. There is very little Arctic sea ice that is over ten years old. (Thanks to global warming, there is very little Arctic sea ice that is over five years old nowadays.)