Are all snow flakes really different?

The short answer to this question is yes -- it is indeed extremely unlikely that two complex snow crystals will look exactly alike. The long answer is a bit more involved -- it depends on exactly how you define "alike." Here are some possibilities:
Case 1: Alike means exactly like. Some things are most definitely exactly alike. For example, our understanding of elementary particle physics indicates that one electron is exactly the same as every other electron. If you think for a bit you will see that this is a profound statement. Electrons are true elementary particles, in that they have no component parts; thus they are all exactly alike.
A water molecule is considerably more complex than an electron, and not all water molecules are exactly alike. If we restrict ourselves to water molecules which contain two ordinary hydrogen atoms and one ordinary 16O atom, then again physics tells us that all such water molecules are exactly alike. However about one molecule out of every 5000 naturally occurring water molecules will contain an atom of deuterium in place of one of the hydrogens, and about one in 500 will contain an atom of 18O instead of the more common 16O. These rogues are not exactly the same as their more common cousins.
Since a typical small snow crystal might contain 1018 water molecules, we see that about 1015 of these molecules will be different from the rest. These unusual molecules will be randomly scattered throughout the snow crystal, giving it a unique design. The probability that two snow crystals would have exactly the same layout of these molecules is very, very, very small. Even with 1024 crystals per year, the odds of it happening within the lifetime of the Universe is essentially zero.
Thus at some very pure level, no two snow crystals are exactly alike. An exception (why does there always have to be an exception?) would be a snow crystal with only a handful of molecules. If we assemble a crystal of only 10 molecules, for example, then it's not so unlikely that each of the 10 will contain two ordinary hydrogen atoms and one ordinary 16O atom. Furthermore, a cluster of only 10 molecules will only have a few likely configurations. So there's a reasonable probability that two 10-molecule snow crystals would be exactly alike.
I might add that even if we restrict ourselves to isotopically pure water molecules, it's still very, very unlikely that two macroscopic snow crystals would be exactly alike. When a crystal grows, the molecules do not stack together with perfect regularity, and a typical snow crystal contains a huge number of crystal dislocations, which again are scattered throughout the crystal in a random fashion. One can then argue, like with the isotopes, that the probability of two crystals growing with exactly the same pattern of dislocations is vanishingly small. Again one has the exception of few-molecule crystals, which can easily be free of dislocations.
Case 2: Alike means crystals that just look alike. Now let's relax our definition of alike, and say that two snow crystals are alike if they just look alike in an optical microscope (the smallest features one can see in an optical microscope are about one micrometer in size, which is about 10000 times larger than an atom). In this case the picture is very different. Snow crystals that fall in Antarctica, for example, are typically fairly simple hexagonal prisms (see Photo Collections), and one can certainly make such simple crystals in the lab. The picture here shows two such crystals grown using the free-fall method (see Free-falling Snow).
Crystals with simple shapes often look similar to one another, and it's not hard to imagine that if you sifted through a reasonable number of Antarctic snow crystals you would find two that were essentially indistinguishable in a microscope. Since simple crystals are very common in the atmosphere (one doesn't notice them much because they're small), it's fair to say that there are a great many natural snow crystals that look pretty much alike.
But that's only for very simple crystals. What makes snow crystal watching interesting is that natural snow crystals often form in beautifully symmetric and very intricate shapes. The combination of order and complexity (i.e. six identical but complex arms) is what makes a well formed dendritic snow star especially beautiful (why we find the combination of order and complexity to be beautiful is another question). The complexity of snow crystal arms arises from the fact that snow crystal growth is extremely sensitive to external conditions (see above). As a snow crystal grows, its external conditions are constantly changing as it falls, and its final shape reflects the time history of these growth conditions. Even a small change in conditions can lead to very different growth. A large beautiful pattern is formed when a snow crystal has a long complex growth history. The more complex the growth history, the more unlikely it becomes that any two crystals will have experienced exactly the same history. And thus it's unlikely to find even two complex snow crystals that look alike in nature.