The Bill of Responsibilities referred to in the question appears to be a list of moral obligations, duties, or responsibilities that responsible citizens of South Africa will undertake. In this case, one could substitute obligations for responsibilities, but as suggested below, responsibilities serves better.
Briefly, the relevant meanings are:
• responsibility: “A duty, obligation or liability for which someone is held accountable”
• duty: “That which one is morally or legally obligated to do”
• obligation: “A social, legal, or moral requirement, duty, contract, or promise that compels someone to follow or avoid a particular course of action”, or “(law) A legal agreement stipulating a specified payment or action...”
Except in contracts, job descriptions, etc., responsibilities are thought of as moral obligations rather than legal obligations. If the qualifier moral is intended but not stated, the word responsibilities is more likely to be used than is obligations; and if the qualifier legal is intended but not stated, vice versa.
In short, obligation has an additional commonly-used legal sense that often is not carried by responsibility or duty.