Both Lean and Six Sigma are process improvement methodologies that seek to add value to the customers, eliminate waste, reduce variation, and ensure long term business success.
There are strong proponents of both Lean and Six Sigma arguing that one is better than the other. There are companies who only work on Lean. They would not touch Six Sigma with a long stick! An example of such a company is Toyota - the originator of Lean concepts. Lean is touted as the enabler for change in culture of the company. Along similar lines, there are consultants who only focus on Lean. At best, they consider Six Sigma as one of the tools within their Lean toolkit.
On the other hand, there are proponents of Six Sigma who claim it is far better than Lean. Some companies have deployed Six Sigma but have not yet jumped onto the Lean bandwagon yet. Most practitioners of Six Sigma have taken the appropriate Lean tools and incorporated into Six Sigma calling the combined toolkit as Lean Six Sigma. The overall methodology is still DMAIC with Lean tools peperred in - roughly 80% of the tools are from Six Sigma and 20% from Lean. GE is one example of a company that initially started off with Six Sigma and then jumped to the Lean Six Sigma journey.
In reality, there are differences in the two approaches that companies need to leverage. Lean alone is not sufficient and Lean Six Sigma alone is also not sufficient. There are Lean concepts (process elements) that cannot be integrated into the Six Sigma project way of working. What is needed is for companies to deploy both: Lean Way of Working and Lean Six Sigma Projects. This combined approach does not have a name yet. Maybe we should start calling it: Lean Sigma.