Are the notions of good and evil different in Christianity and Islam?

Yes, they are absolutely different. In Christianity God is the absolute moral law giver and final authority. In Islam, Allah, is the absolute moral authority and law giver. If you read the Bible and the Quran you may find little overlap in some of the moral codes but will certainly find differences.

I wouldn't summarize the entire moral code in the bible by the golden rule. The moral code of the Bible is better summarized by the greatest commandment:

36 “Teacher, which is the great commandment in the Law?” 37 And He said to him, “‘You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.’ 38 This is the great and foremost commandment. 39 The second is like it, ‘You shall love your neighbor as yourself.

You can take all of the 10 commandments and much more and fit it into these two categories.

This is from the New Catholic Encyclopedia (2003) entry on Islamic Law (sharia):

Islam is a lay theocracy. God is head of the Islamic community. God alone legislates, and His will is carried out by the community of believers. The law is what God has made known to the community of believers through revelation to the Prophets (among whom are Moses and Jesus), the last of whom is Muhammad, ‘‘seal of the Prophets.’’ The prophets are the obedient instruments of the revelation, passing it on intact to the believing community. In its broadest sense, the law encompasses articles of belief, regulating man’s relation to God (religion), as well as rights and obligations regulating man’s relation to his fellow man (law).

The law is found in scripture and tradition; i.e., in the Quran, which is the sacred book of Islam, and in the Tradition of the Prophet Muhammad. God’s revelation ceased with the death of the Prophet.

A Muslim submits to the law as a precept of faith (the literal meaning of Islām is ‘‘submission’’); in so doing he fulfills his duty toward God. ... Disobedience to the law is a sin committed against God, not only an infringement of the legal order.

Liberty is the starting point of the law. This is illustrated by the five categories under which the actions and omissions of men are classified: (1) obligatory, (2) recommended, (3) permissible, (4) disapproved, (5) forbidden.

But liberty cannot be absolute, lest it destroy itself. So the function of the law is to set limits on this liberty so as to guide it towards the benefit of the individual and society. Good and evil are defined by the law, that is, by God. It is divine voluntarism that characterizes Islamic law; there is no question of natural law as a ‘‘supreme reason existing in God,’’ inscribed in the very nature of things. In Islam, there is no question of positive law subordinating itself to divine law and respecting its dictates; they are both the same: Islamic positive law is a divine positive law.

The study of the law became the most respected of all endeavors, assuming a position commensurate with its object: it enabled the community to fulfill the divine precept (Qur’ān 3:106) that it ‘‘promote the good and repress evil.’’

Meanwhile, regarding the Catholic doctrine of evil, the same encyclopedia states:

‘‘Evil’’ can be defined as that which opposes, or is the antithesis of, what is good. There is no precise articulation of the nature of evil in the creeds of the Church, nor is there any explicit or definitive Christian doctrine of evil.

Traditionally, Christian theologians have referred to the Adamic Fall (Gn 3) for an understanding of the source of evil, the view that evil emerges from the misuse of human freedom. While this had been foreseen by God, the gift of freedom, nonetheless, was an essential and fundamental gift from God, one that distinguishes humanity from all other creatures and gives man alone the ability to choose good and evil.

A common division of evil is that into metaphysical, physical, and moral. ... In contrast to physical evil, moral evil is that found in a rational and free nature as such. Properly the soul is its immediate subject, or more precisely, the will, with its power of obeying or disobeying the norms of moral conscience and the divine law. Moral evil is therefore a privation of rectitude required by the natural law, a privation affecting a free will, which through its own fault lacks a perfection it ought to have (De malo 2.1–2). ... [M]oral evil results from the voluntary activity of an agent who, in depriving himself of a perfection to which he is obliged by nature, inflicts upon himself a self-mutilation.

With these two texts in mind, and declaring outright my highly incomplete understanding of the topics at hand, there seems to be significant parallels between the two. Both see evil as an act of the will against God. The key difference is in the different degree of codification of these acts into law. Islam seems to have a greater number of religious laws defining evil acts than Christianity. As such, for Islam, the codification seems to be much more "rigid", up to the extent that the legal system must be fully defined by the religious law found in the Quran (and why countries like Saudi Arabia, Iran and so on have legal laws strictly founded in religious law, like theocracies). Notice the same might be the case for some forms of Judaism. Conversely, in Christianity, there is no strict codification of evil acts to the extent of merging these with legal codes (even if this might have been the case in some Christian kingdoms in the past). Similarly, the list of things to be forbidden, derived directly from some written law, seem to be much less than that in Islam.