Those concepts are usually mixed everywhere. It seems both are based on Lean, but that's all. The core of the Lean Startup aproach seems to be to develop the MVP (minimum viable product). Whereas the core of the Lean software development seems to be to eliminate waste.
Could we say that?
I would suggest that Lean software development as discussed at length by Mary Poppendieck and Tom Poppendieck is a methodology closely related to the Agile family of software development methodologies such as Extreme Programming and Scrum, each with its own design emphasis, rituals and community. However, at the heart of all Lean/Agile culture is the core belief that rapid feedback cycles - where feedback is meant to be in the form of ‘live’ reactions (such as sales or usage spikes) from the ‘real environment’ (from clients, prospects and users most importantly) – are required to discover the important but false assumptions often hidden in planning or requirements documents. So you can think of the ecosystem of Lean/Agile practices as vehicles for enabling fast feedback cycles - ‘fail fast to succeed sooner’ – in order to expose and manage the risk associated with false assumptions about the development environment.
Lean Startup simply applies the idea of using rapid feedback cycles to expose false assumptions to the customer development and business side of the organization. Steve Blank coined the term and created the processes around ‘Customer Development’ which Eric Reis built upon to develop ‘Learn Startup’.
Kanban is a tool, like you mentioned, that can be used in any project management style, even Waterfall.
I think the following couple of articles are perfect to understand the difference easily.
From oxzigen.com - lean software development:
It’s easy to confuse the Lean IT and Lean Startup methodologies. (...) I sometimes use the analogy of a restaurant to describe these different roles. According to this analogy, the Lean Startup activities are taking place mostly in the “dining room” where there is direct contact with the customers. (...) Lean IT activities, on the other hand, are taking place mostly in the “kitchen.” Their goal is to deliver a high-quality product quickly and efficiently.
From w2lessons.com - minimum viable product dissected:
One of the fundamental and most misunderstood tenants of the lean startup is the minimum viable product (MVP). (...) For starters you must look at the MVP not as a product but as an experiment. (...) The fact is that the actual product does not have to exist at all. (...) Rather than build a whole system, build a simple landing page with some screenshots from your graphic designer (think of about.me).
