No Snowflakes Alike?

Prompted by this Citizen Dog: I've long doubted this factoid. First of all, how many snowflakes are there in 1 snowfall on 1 day in 1 square mile? Millions? In the world at this second, there must be billions or even trillions of snowflakes. Today alone. And of those snowflakes none are 'alike'? Well what level of similarity are we restricting here? This seems to render the definition of 'alike' useless.

The wiki quotes the claim that:

It is next to impossible that two snowflakes are exactly alike due to the roughly 10¹⁸ water molecules which make up a snowflake, which grow at different rates and in different patterns depending on the changing temperature and humidity within the atmosphere that the snowflake falls through on its way to the ground united.

So thank goodness, the resident genius of the Straight Dope is on the case: whose answer demonstrates that we think so much alike that I should declare here and now that we are not the same person:

Chances are, in fact, that there are lots of duplicates. What the snowologists really mean is that your chance of finding duplicates is virtually zero. It's been calculated that in a volume of snow two feet square by ten inches deep there are roughly one million flakes. Multiply that by the millions of square miles that are covered by snow each year (nearly one fourth of the earth's land surface), and then multiply that by the billions of winters that have occurred since the dawn of time, and it's obvious we're talking unimaginable googols of flakes. Some of these are surely repeats.

See!

Second pic from here.