Economics and charity
Onto the economics. Questions of "should charity be provided by the government or individuals" aside, you can ask whether charitable contributions should be tax deductible. Classical economics would be pretty comfortable with that because of the externalities in charity -- for not only does your charitable contribution do you good (because it makes you feel better), it presumably also helps the person receiving the contribution. But since you don't really feel the "happiness" experienced by the recipient, you end up not giving enough. A tax deduction lowers the price of charitable giving and thus increases its quantity, getting us closer to the appropriate amount. (The trade off is higher marginal taxes which slow economic growth and reduce wealth overall).
Not feeling the "happiness" experienced by the charity's recipients also makes us end up giving to sub-optimal causes, and so reduces the quality of charitable giving. Since we don't know how effective a non-profit is at using the money, we may not support the most productive ones, and we may also give to things we feel are important to us, but may not be that valuable to the poor recipient.
Chicago's Judge Posner argues that nonprofits' operating constraints (which, among other things, cap salaries and expenses as a proportion of revenue) are designed to tackle some of the quality problems. By limiting the organization's ability to make profits and limiting salaries, it will be forced to maximize output, which is the least worst we can hope for since it's much easier to track how much "good" a charity is doing than how "good" that "good" is. And if you go to your average non-profit, you'd see lots of stats measuring outputs, but very little measuring "outcomes".
This focus on output over outcome creates its own wacky pathologies that are really really hard to fix (I've worked in this field, take my word for it). Giving allocation control to those who use the services, by giving poor people cash, may improve their wellbeing, but it reduces their incentive to work, and society is uncomfortable with that type of unrestricted redistribution (except where taxes are involved). Anyway, go read Jane's post, where she discusses exactly these issues at greater length.