Usually there are different roles in the organizations for digital transformation, some organizations have two department for digitalization. One the IT department and the other is digital transformation department. What is different and what is the role and responsibility of each? How do you avoid overlapping yet empower innovation in both? Especially interested if talking about a company in the oil and gas industry.

IT Team is majorly occupied in BAU but Transformation team will focus on bringing the change in the organisation for users , data centralisation & accessibility with more governance.

Digital transformation in the IT department usually focuses on providing technology or leveraging current technology to enhance business process whilst the other has a broader scope and focuses on how the business can change its business operating models, processes, organisational structure and culture to best utilise technology as an enabler to meet business objectives. Both areas should foster a culture of innovation and change, the second tends to review look more holistically at what will enable the organisation and its people to meet long term objectives and build internal capability. It is sometime in the strategic planning or corporate area and referred to as Business Transformation. 

I think you talking about the "standard" digitalization, i.e. turning physical things into digital making it available by an app/system, having a well know trail. And the other is the digital innovation, which I think is harder to do, because gathers business and digital knowledge in the same conecepts, needs a multipurpose team and requires more rethinking than the first one.