If you help me because you want Allah to give you rewards or presents , then there's nothing special about your kindness , isn't it?. I would say that an Atheist helping other people is BETTER than a Moslem helping other people because they want Allah to give him 'presents' in returns (sorry to say this). There is no sincere thing on this planet. Would believers still help other people if Allah were never promise them anything, Allah never punishes them, and heaven and hell don't exist??
The premises of your question are objectionable.
Religious people may give out for Divine reward, but they also give out because religion teaches them to practice compassion and charity. Religion tells them that they make up one family descending from Adam and Eve, and together they are like brothers and sisters; that they must be concerned for one another. Islam tells us that we are all created from "one self" and all have a share of Divine spirit in us. So this ideally gives Muslims a sense of community, connection and care. And like every community system, there are rewards for cooperation.
A deeper analysis tells us that man is never truly and fully selfless, simply because the self is always with us. Islam talks about different types of nafs and that the nafs will finally taste the consequences of his deeds. Therefore the choice is between following your lower nafs or higher nafs, your evil nafs or your angelic nafs. There's no other alternative.
The higher nafs, due to having a greater share of the Divine spirit that is shared by all of us, inspires deeds that have greater benefit than the immediate benefit to the individual doing the deed, because the Divine spirit is connected to all individuals. But once we follow our base desires that are more attached to our bodies, we move further from this spiritual unity and embrace greater individuality. So it is recognition of spiritual unity that engenders compassion and charity in us, and heavenly rewards are only manifestations of this greater consciousness which we call "faith" and good deeds are consequences of this faith.
An atheist may engage in good deeds but if he is truly doing it without expecting tangible reward (which is questionable), he must be doing it for recognition of some unity and connection among human individuals and developing a charitable feeling as a result. This charitable feeling can be said is itself the reward of the charitable deed. When one feels generous, it is due to compassion that one feels in his heart. With compassion indeed comes a feeling of pleasure, "a feel good" experience which acts as both the motivation and reward for acts of generosity. When you understand this, you understand that even an atheist's charity is not truly selfless. There's a good feeling and an urgency to relieve one's self from a pressure on one's conscience that motivates the good deed.
Now Islam tells us this sense of connection and compassion that motivates us to generosity is in fact a spiritual connection. An atheist may not agree, but again the atheist's admission or non-admission doesn't change the fact.
However, because a believer's good deed is based on this deep consciousness of our spiritual unity, he is far more likely than an atheist to spend in charity, because this consciousness itself invites greater share of the Divine spirit and mercy to the believer that is in turn passed around by the believer to others.
This I think is what ideally runs beneath a Muslim act's of charity. But I understand most Muslims tend to think of charity as a "divine decree" that we must just fulfill to attract Divine reward, which gives the wrong impression that an atheist doing good deeds is doing it without expecting reward. This is unfortunately the consequence of the legalistic reductionism of religion that is prevalent in most of the Muslim world, but if we recognize that reward is just the consequence of greater participation in the Divine mercy and spirit, and that this sense of mercy and spirit is immediately felt and present in our good deeds, to greater or lesser degree, giving us a sense of pleasure and satisfaction, then it is impossible to argue there can truly be no expectation of reward with acts of generosity.