What are and differences and the similarities between social media managers, community managers, and digital marketing managers?

Very broadly speaking, and with much generalisation:Digital Marketing Managers develop strategies to use the Internet to promote the brand. They generally have a broad understanding of the opportunities that the various online tools and platforms can afford.Social Media Managers work out the tactics required to make that strategy happen for the brand. They have a deep understanding of how to make the most of the various online tools and platforms available and what's required to make those tactics work.In the context of the other two roles, the Community Managers "implement" the tactics, but it's more complicated than that.Community Managers can immerse themselves into the communities that are being reached out to, they can detect and understand the subtle dynamics that are present in online communities. A community manager has to be two people: they represent the brand to the community and they represent the community to the brand. So, it's more than just implementation.

All social media managers are community managers but not all community managers are social media managers.

In very cut short terms, a social media manager develops and revises the social media monitoring, listening and engagement strategy. They analyze and create all the branding on social media sites in accordance with the marketing team.

Community managers are the brand ambassadors and implement the social media strategy of their brands set by digital media managers and social media managers. They tend to become experts on online sales and reputation management. Today, leading community managers are grooming themselves to practice social experience management. 

The digital media manager role is slowly taking over the role of the social media manager because it is broader. They go beyond social media and also develop and delegate and play an advisory role to the marketing team. Besides social media strategy they create a plan for:

Inbound marketing (accompanied with newsletters, webinars and online customer support)

Work with marketing and IT team to work on a customer retention and acquisition startegy.

Online lead generation

Viral marketing using video

SEO, SEM, Facebook, Twitter and Linkedin Ads

Very important decision they carry out is media buying and selling. 

Social media mananger operates from the edge of the company, managing brand recognition outside of thr scope of the brand website wellas Community manager operates within the company , managing customer relationships with a brand or product and digital marketing manager promotes a brand or product online using all available digital channels. Their similarity is that they all work to promote a brand.