#Octave, 179 bytes
A=' *';C=D=E=zeros(25,80);B=[i=1,1,(1:78)*0];do if i++>25 B=B&0;end
C=[B(randperm(80));C(1:24,:)];D=[B(randperm(80));D(1:24,:)];F=D;D=E;E=F;pause(.1);disp(A((F|=C|D)+1))
until(~F)
##Ungolfed
A=' *'; % Index with 1,2 to translate to ' ','*'
% A([2 1 2 2 1]) => '* ** '
C=D=E=zeros(25,80); % Locations of fast snowflakes,
% slow ones moving on frame 1,3,5...
% and slow ones moving on 2,4,6...
B=[1,1,zeros(1,78)]; % Line with two snowflakes in it.
% Will be added on top each frame.
i=1;
do
if i++>25 % We push 4 snowflakes per iteration.
B=B&0; % Stop new input after 25 iterations.
end
C=[B(randperm(80));C(1:24,:)]; % Shift fast snowflakes down
% and add randomly permuted input line.
D=[B(randperm(80));D(1:24,:)]; % Shift one set of slow snowflakes down
% and add randomly permuted input line.
F=D;D=E;E=F; % Swap the two sets of slow snowflakes.
pause(.1);
disp(A((C|D|E)+1)) % Print '*' where either of the matrices
% is set, ' ' otherwise.
until(~(C|D|E)) % Loop until the matrices are all zero