Both concepts are vague and often used interchangeably, and their dictionary definitions are almost indistinguishable:"the state or fact of being accountable or to blame for something" vs. "the state or quality of being dedicated to a cause, activity". Yet, there is a touch of a more passive voice in the first than in the second, and this distinction seems to be sharpened by some philosophers. This impression is confirmed by del Corral's analysis in Commitment and Responsibility:
"Responsibility is a very broad concept. Moral and legal philosophy have provided many different definitions of what responsibility is, as well as different set of criteria an agent has to meet in order to be held responsible. The claim I will defend here is that responsibility can be understood in a non-moral and non-legal sense, as a relation between an agent and an outcome to another agent, who attributes responsibility. This relation is basically explanatory: an agent is responsible for an outcome if it is possible to explain the outcome in terms of the agent's authorship."
To put an emphasis on it, one makes a committment, oneself, but is held responsible, by others. Of course, one can hold oneself responsible too, but usually after the event, and there is an air to it of treating yourself "in the third person", as it were. Del Corral also discusses the responsibility-for-omission problem that you mention:
"Agents are often held responsible for something they have not done, such as preventing an outcome. Omissions are problematic because the relation between the agent and the outcome is not evident: there has to be a link of some kind that connects agents with outcomes. However, it is not clear whether that link is necessarily causal, or on the contrary, causal effectiveness is not necessary for responsibility."
This is only one of several problems with demanding a causal link between agents and what they are responsible for. See also the IEP article on Responsibility:
"The responsible person can be relied on to judge and to act in certain morally desirable ways; in the case of more demanding ("more responsible") roles, the person can be trusted to exercise initiative and to demonstrate commitment; and when things go wrong, such a person will be prepared to take responsibility for dealing with things." [emphasis mine]
Again, there is this active/passive duality between "demonstrating" commitment and "taking" responsibility, "initiative" and "dealing". According to Clemence's Existentialism: A Philosophy of Commitment this active aspect of commitment assumes even grander proportions in existentialism:
"Commitment can mean many things: a promise to keep, a sense of dedication that transcends all other considerations, an unswerving allegiance to a given point of view. In existentialism, commitment means even more: a willingness to live fully one's own life, to make that life meaningful through acceptance of, rather than detachment from, all that it may hold of both joy and sorrow."
Brandom, a leading contemporary pragmatist, made commitments (and entitlements) in the "game of giving and asking for reasons" into the central concepts of his philosophy of meaning, language and social behavior. They serve as bases for "scorekeeping" by various participants in the game on other participants, thus enforcing the normative use of language and reasoning that makes them meaningful. Brandom's sense is more specialized, but it also emphasizes the active aspect, see Standefer's blog on Making It Explicit:
"Brandom tells us that the primary normative concept for inferential articulation is commitment. When we move to the social picture involving more than one agent, there is a shift to multiple primary concepts. They are commitment and entitlement... Commitments have a sort of double life. Not only are they undertaken, but they are also what one is entitled to. To put it awkwardly, one can be committed and entitled to commitments... Brandom should probably have said that the fundamental normative status for the game of giving and asking for reasons is entitlement, and commitment takes on its content role."