Saudi scholars claim that it is against the Sharia for women to drive. What are their evidences and reasoning?
Basically, it was their opinion that it could lead to other haram actions: things like travelling without mahrams, mixing with the opposite gender, abandoning of Hijab etc.
Even the actual head of the Council of Senior Scholars Abdul-Aziz ibn Abdullah Al ash-Sheikh an offspring of Muhammad ibn 'Abdalwahhab is known to be against this as he considers that this might lead to haram acts and big malicious things. While he speaks about weak or seducible people (men more exactly) whom might be lead to these haram acts. So I don't think he changed his mind about that.
As possible evidences he quotes the fact that according shari'a a woman shouldn't travel alone or receive any foreigner in her house without the presence of a mahram and ordered to veil her. (here an article in Arabic)
One could add that leaving the house without the guardians permission is also not allowed. "And who knows what these "free driving" women will do and where they might go with a car"!
All this can basically be regarded as maslahah (public interest) or sadd a-dhrai' (closing a halal "door" which may lead to haram) in the osol al-fiqh (see for example What are basic (osol al fiqh) differences & similarities between the 4 (existing) Legal Sunni Schools: Hanafi, Hanbali, Maliki, & Shafi'i? Charts?).
Note that already in 2011 sheikh 'Aid al-Qarni has been quoted to say that basically that prohibiting women to drive in Saudi Arabia has no backup nor clear evidence in the shari'a texts. This was in a TV-Interview with al-Arabiya Channel after the arrest of the women's rights activist Manal al-Sharif in May 2011. He also quoted the example of the polemic about the education of girls which was launched in the kingdom in the era of king Faisal to compare it with the polemics about women's right to drive a car. (For details read the article in Arabic here)
The major reason for Saudi Arabia to allow women to drive is politically driven and initiated by the U.A.E. (who have big influence on Saudi Arabian politics actually as the strongest economy among the Gulf Cooperation Council states (after "the expulsion" of Qatar)) who wants to present the actual crown prince and future king as a moderate and open minded Muslim leader.