How does sulbutiamine work in the same way as thiamine, even if it doesn't have a thiazole ring?
To get started, here are the structures of some compounds that are relevant.

Thiamine



Sulbutiamine



Allithiamine



Looking at these structures we see that you are of course correct, sulbutiamine does not contain a 5-membered thiazole ring like thiamine.  

Thiamine (vitamin B1) was an early focus of interest because sailors at sea for long periods of time developed beriberi due to a nutritional thiamine deficiency.  Since those early days, thiamine substitiutes were of medicinal interest.  One of the early substitutes was allithiamine.  It is a natural product that can be extracted from garlic.

You can see the structural similarities between allithiamine and sulbutiamine.  Sulbutiamine has the advantage that both sides of the disulfide (S-S) bond contain the same active ingredient, so one would expect increased pharmacological properties from sulbutiamine.  The advantage that both allithiamine and sulbutiamine possess over thiamine is that they are more soluble in lipids (the positively charged thiazole ring in thiamine makes it very soluble in water, but not so soluble in lipids).  This enhanced solubility in lipids means that they can cross the blood-brain barrier much more effectively than thiamine, this improves their bioavalabilty.

With that background we can now answer your question, "how does sulbutiamine work in the same way as thiamine."  

Once in the brain, sulbutiamine can be easily metabolized to thiamine.  Look at the structure of sulbutiamine and here is what happens:


reductively break the disulfide bond and form two identical mercaptans
$$\ce{RS-SR -&gt;C[{[H]}]\   2 ~RS-H}$$
protonate the acetyl (that's on the nitrogen that will become part of our thiazole ring) oxygen to make it more attractive to nucleophilic substitution
Let the mercaptan sulfur nucleophilically attack the acetyl carbonyl carbon
eliminate water and form the 5-membered thiazole ring
and we now have thiamine


But a much larger dosage in the brain than had we simply ingested thiamine.