The advantages of allowing charities to campaign politically, is to protect the organizations freedom of speech, and protect the IRS from selective enforcement. The disadvantage, is that the charity is usually granted tax exempt status, and donations to a charity are usually tax deductible.
For example, in the USA, a charity is a 501(c)(3) organization:
"which does not participate in, or intervene in (including the publishing or distrbuting of statements), any political campaign on behalf of (or in opposition to) any candidate for public office."
This prohibition has made churches reluctant to make statements regarding issues that normally be within a churches teaching, for example the Catholic churches' stance on abortion. There is also the famous case of Billy Graham who enacted an advertising campaign to support candidates that endorce "the traditional definition of marriage." This was viewed as advocating for Mitt Romney.
Some churches have advocated civil disobediance, directly defying the law which they see could be used to selectively attack organizations with differing views than the government. In the definitive case Branch Ministries Inc. v. Rossotti, the court found:
The government has a compelling interest in maintaining the integrity of the tax system and in not subsidizing partisan political activity, and Section 501(c)(3) is the least restrictive means of accomplishing that purpose.
Integrity is very important to the IRS, especially considering the recent targeting of Tea Party 501(c)(4) organizations.