汉字和拼音文字相比，有什么优势？

When you write a text message or a twitter with a limit of word-count, you'll find Chinese character is better than Roman alphabet.

To write a word, if the reader knows what I'm going to write, the information he gets is zero. I need not to write, because he already knows.
If the information has two alternatives Y or N, then the reader sees you write a Y. He didn't know what I'm going to write, but he knew there's a 50% probability. Now he knows, so the price of the information is 50%.
If more alternatives are included, the choice of one out of a rich variety is of course more significant than one out of two.
In the case of the amount of information, since the more alternatives there are, the less likely each alternative is likely to occur and the more its occurrence will mean, one might say that the amount of information is inversely proportional to the probability.

If I'm going to write a letter of English, there are 26 alternatives. (even if 26^2 is 676)
If I write a Chinese character, there are more than 2500 alternatives. (commonly used 2500 characters)
So when the amount is fixed, a Chinese character gives us more information.
We pay a high price to learn more symbols for more information.

So, to write Chinese character is to save paper and drive capacity.

In the old time, writing materials were very heavy, so people tried to write with economy words in order to write more information on one piece of material. Chinese characters can transmit more information in the same space.

There're 17 symbols on this material. The transcription is:

PERIKLES
XSANΘIPPO

What did it tell us? It's a name, Perikles Xanthippos. Yes, the famous politician of Athens. The second line is the name of his father, taking the part as his family name. So, just a name, in 17 symbols.

(Today Chinese readers still complain about the long Western names, for example when they reading One Hundred Years of Solitude)

Then look at a material with 13 symbols. The transcription is (from top to bottom, RIGHT to left):

(1) ......不雨
(2) ......其雨 之夕允
(3) ......今夕其雨
(4) ......隻象

The translation is:

(1) ......not rain

(2) ......that rain; till dusk permit

(3) ......today dusk that rain

(4) ......hunt elephant

It told us a lot of information with only 13 symbols.

Here's an example just in Modern Chinese. The text in Chinese characters takes less space than that in phoneme writing system(Shortened Pinyin).
深藍的天空中掛著一輪金黃的圓月，下面是海邊的沙地，都種著一望無際的碧綠的西瓜，其間有一個十一二歲的少年，項帶銀圈，手捏一柄鋼叉，向一匹猹盡力的刺去，那猹卻將身一扭，反從他的胯下逃走了。
Ŝēnlán de tiānkōŋ ẑōŋ ɡuà ẑe yìlún jīnhuáŋ de yuányuè，xiàmiɑn ŝì hǎibiān de ŝādì，dōu ẑòŋẑe yíwàŋwújì de bìlǜ de xīɡuā，qíjiān yǒu yíɡè ŝíyīèrsuì de ŝàonián，xiàŋ dài yínquān，ŝǒu niē yìbǐŋ ɡāŋĉā，xiàŋ yìpǐ ĉá jìnlì de cìqù，nà ĉá què jiāŋ ŝēn yìniǔ，fǎn cóŋ tā de kuàxià táozǒu le。

The phoneme is according to Hanyu Pinyin(1958). Here ẑ ĉ ŝ is short for zh ch sh, and ŋ is short for ng. So each phoneme can be written in one letter. The phoneme of tone do not take extra space, and they are written as superscript.

Please feel free to help me with my grammar problems in my answers.