Your bow is very much responsible for the tone that ultimately comes out of your bass. Each bow has tonal and response characteristics all its own and each bow will sound very different from the next.
Even bows from the same maker won't sound exactly the same as each other. When I bought my last bow, I had the bow maker send me 3 bows to choose from, and each was unique. Also, another thing to remember is that a given bow may sound better on a certain bass and not as good on another. You usually have to consider how a particular bow sounds with YOUR instrument.
Certainly you should notice a great improvement in tone from a fiberglass bow to a wooden bow, particularly if you buy a bow made of pernambuco (the wood of choice for fine bows). Brazilwood is generally a good step up from fiberglass but pernambuco is the way to go if you can, however expect to pay at least $1000 for a pernambuco bow (although most are quite a bit more than that).
I guess it all depends what your needs are. For me, I'd almost rather have a great bow than a great bass. It's that important to me. I can generally make most moderately good basses sound the way I want with my favorite bow, but I'd have a harder time making a fine Amati bass sound great with a cheap bow.
My advice, if you are interested in playing arco, is to get the best bow you can afford.
