Why there are two versions of the History of Tang? What about the Old History of Tang that offend the Song emperor?
Is the New History of Tang better than the old one?

It was not at all uncommon for many more than one dynastic history to be compiled. During the Tang dynasty itself for instance, a new history of the Qin dynasty was compiled (in 644-6) and presented to Taizong despite the fact that there were at least some 18 extant histories of the Qin already in circulation.

Of course the compilation of dynastic records also became much more routinised under the Tang with the establishment of the Shiguan or Bureau of Historical Records in 629. Generally, this set-up meant that the writing of dynastic histories was (in theory) an autonomous scholarly undertaking that was not unduly influenced by political pressures (again in theory). So I would not look first to the direct agency of the Song emperors as the main or only reason for writing a new Tang history.

More likely I think the reason has to do with a general shift in outlook shared by court and court officials at this time. The Neo-Confucianist ethos of the Song influenced aspects of the redaction of the Xin Tangshu which recast received material to reflect a more moralising set of historical exemplars. So perhaps the increasing influence of Neo-Confucianism was one of the main reasons for wanting a more updated and "relevant" version of Tang history?

Then why is only History of Tang and History of Five dynasties were rewritten? Why didn't they rewritten more history books to better fit the changing political outlook?

Good question. One simple answer may that these were the immediate predecessors to the Song. It's reasonable that Song scholars would have the greatest interest in these preceding periods. There might therefore even have been the ability to collect and input additional source material that had circulated outside of the Jiu Tang Shu. There would have been many specific reasons I imagine for Song Neo-Confucianists to want to revisit these periods such as concerns regarding the high status of non-Han Chinese during the Tang, the important role of the military elite, and the history of relations with non-Chinese peoples.

I have not compared the two histories with these particular questions in mind. Someone more knowledgeable should be able to point you to historiographical studies comparing the two works at any rate. If this has not yet been done (and if has perhaps the treatment is only in Chinese or Japanese) it should give someone an excellent doctoral thesis topic.

The following article does not deal with the specific question you posed but throws interesting light on the contexts for history-writing and historical revisionism during the Song:

Robert M. Hartwell, "Historical Analogism, Public Policy, and Social Science in Eleventh- and Twelfth-Century China," American Historical Review 76 (1971), pp. 690-727.

It underscores the importance of (Neo-)Confucianist moral didacticism as a main driving force in historiography during this period. Previous historical periods were examined for moral exemplars more than anything else.

Agency for this came from among the intelligentsia it seems rather than from the throne. The intended audience was two-fold:

-the emperor. Some contemporary works "distilled" key moments for illustrating moral lessions and as such served as the mirror of princes (or advice to the ruler much like the western medieval genre of speculum principis). The (re)interpretation of the Tang dynasty was esp. crucial since Song officials insisted that Song institutions derived from those of the Tang in important ways.

-future scholar-officials. The civil service examination system included examination of the history of preceding dynasties and relied on specific interpretations set in dynastic histories. Having a new Tang history, for instance, that stress the role of morality etc. would reshape the way whole generations of Song scholar officials regard the past.