I assume that you are asking about Guns, Germs and Steel because you tagged your question with that and because Diamond talks about writing systems in his book. I have edited your question and moved it to reflect that.
The first answer does not refer to what Diamond is talking about and is also factually incorrect. It talks only about the Chinese writing system whereas Diamond uses the term "Eurasian" to mean anything on the continents of Europe and Asia. So both Chinese characters and the Greek alphabet are Eurasian in his terms. As you can see on p. 217 of the book (early Chapter 12), not all Eurasian languages used alphabets, so that is not really a difference between Eurasian languages as a group and others.
The only thing I know of in Diamond's book that suggests that writing systems developed differently can be found on p. 360 (middle of Chapter 18). There, Diamond says that Eurasian civilizations tended to have relatively large groups of people who were literate. This meant that they had an advantage over the people of the rest of the world. In the rest of the world, (Mesoamerica, for example) only a small elite were literate, he says.
Diamond says, then, that the major difference was that writing became more widespread in Eurasia than it did elsewhere. This gave Eurasians an advantage because their population was more educated and more able to innovate because of it.
As you can see at the PBS link, the Eurasians' advantage was partly due to their geography which allowed writing systems to spread easily (just as crops were able to spread more easily in Eurasia, according to Diamond).
Two of the most distinct differences include what the characters in a language represent and the order in which something is written.
For example, for each sound in a word, we represent that with a letter. We have a 26 letter alphabet with even more sounds. Eurasian languages have characters that are essentially equal to words for us.
We read from left to right and top to bottom. Their writings are exactly the opposite, bottom to top, and right to left. All of this information I know from studying Chinese culture for the Academic Decathlon of 2007.
Maybe another editor can give you a reference with a link for you to read.