The difference between simplified and traditional Chinese and how should one convert a traditional sentence into a simplified one?
Now how can it be converted to a pure mainland Chinese version?
It's a big project. Best to find a professional translator who knows both well to do the job.
I mean are the differences just in typing?
No
Are there any difference in the vocabulary?
Yes. Quite a lot are different. Like software, in Taiwan it's 軟體 and in the mainland it's 软件. I'd like to show another example, the word cache in computer science. In Taiwan they adopt 快取 while in the mainland 缓存 is used. The two versions seem quite different at the first glance. One uses 快 fast, the other 缓 slow. But they describe the very same thing from two aspects. Both 快取 in simplified and 緩存 in traditional will be odd.
Is there any service on the net which gets a traditional word (or a word typed in traditional Chinese) and then gives you its simplified version?
Character-to-character is available, like from 體 to 体. But word-to-word is not that easy. 软体 is improper. Besides, there are cases that multiple traditional characters can be transformed into one simplified one.
Do I have to apply grammatical changes, too?
Grammars of both are almost the same in the written language, but I'm not sure about it.
Do simplified Chinese (People who are living a mainland China) understand traditional Chinese easily? What about vise versa?
The written language can be easily understood. The characters can be recognized easily though one may find it hard to write them by hand. I know little about the case vise versa since I'm from the mainland. Traditional characters are never forbidden though they cannot be used in the official documents like laws. We learn to recognize the traditional ones but not how to write in the school. I have read a lot of novels printed in traditional Chinese. Both of them can be widely seen in daily life, of course the simplified more than the traditional. Anyway both are precious legacies from our ancestors of the past thousands of years.
Which one is used wider in the world (traditional or Chinese)? (I mean for promotional purposes is it better to translate a passage into simplified or traditional to be understandable by more people?)
So far as I know, many softwares, webs and manuals provide both. But I myself find it okay to read only the traditional version.