"Human touch" or "humane touch"

Here’s what ‘The Cambridge Guide to English Usage’ says:

There are loftier principles in humane, and a humane approach . . . connotes compassion and concern in situations where others might react harshly. The reactions implied in human are much more down-to-earth . . .

As for your particular instance, human touch is the more ususal expression. Touch collocates more frequently with human than with humane. In the Corpus of Contemporary American English there are 122 records of human touch against two for humane touch. The British National Corpus figures are 19 and 0.

The noun human refers to a person. As an adjective, human means showing the distinctive characteristics (good or bad) of people, as distinguished from animals. The adjective humane means characterized by kindness, compassion, or sympathy.

For Example :
* Ram is human. (= You are a person.)
* Ram is humane. (= You are kind and merciful to people and animals.)

Barrie England's answer has already given a bit of the distinction between human and humane, but I think it's important to expand on this. "Human touch" is an idiom, meaning a friendly, and pleasant way of treating people, or having the quality of being friendly, etc.

So a workplace might lack a human touch, in that it is not an inviting or friendly place to work. A person might not use a human touch when dealing with others, in that they are cold and unfeeling or unempathetic. So when you use the human touch you make others feel welcome.

"Humane touch" is not an idiom, however, it alludes to the "human touch" idiom. When something is humane, it means

characterized by tenderness, compassion, and sympathy for people and animals, especially for the suffering or distressed

So for a charity, using "Humane touch" as a tag-line is saying that this charity's touch is characterized by tenderness and compassion. As a phrase it sounds off to me because it seems like a mistaken use of the well-known idiom. But as a slogan it makes sense, because it refers to the original idiom while emphasizing caring and compassion over friendliness.

My personal opinion is that this tag-line is a little too cliche sounding, but its meaning is clear and (depending on what this charity DOES) it sounds like a reasonable tag-line.