How thick is a snowflake?
There are many different types of crystals and their fromation is far too difficult for me to understand. This answer simply deals with the thickness part and that too merely gives a rough approximation due to the lack of data.
 From CalTech

   

We have two formulae for calculating the width:  


For Dendrite
$$\mathrm{h =a_2 D^f} $$
D: crystal diameter (cm) (major dimension)
h: crystal thickness (cm) (minor dimension)  
For Needles
$$\mathrm{d = a_2 L^f} $$
D: crystal length (cm) (major dimension)
d: crystal thickness (cm) (minor dimension)    


This is a table for $\mathrm{a_2, f}$ 

  

For d and l   (right most column )
   

I calculated the thickness of a $\pu{3 mm}$ Dendrite which is coming around $\pu{24  \mu m}$. 



Source  

https://web.archive.org/web/20180416083128/https://www.nsstc.uah.edu/data/uahsevere/owles/parsivel/references/Snowflake_Size_Distributions_Lecture.pdf