To clear your doubt first of all you should be clear about What is Social Media and What is Community?

Social Media is a Website of an App which permits it's users to share and post information in form of video, picture or content and add/follow friends, colleges or other users  which means participating in Social Networking Site.
Role: Social Media Manager is the one who manages all social media channels for promotion of product by developing strategy and by allocate goals to the team or to his own for increasing product/brand awareness with a goal to get more positive engagement. 


Community refers to the group of users/people who have common interest or shares the same interest of information, view, values, business goals. We can also call it Social Community which includes a group of like-minded people connected to share the same taste of knowledge and interest.
Role: The community Manager role is to manage the community by guiding the subscribers how and what activity to do on the basis of the community interest. He also create the rules and regulation/guidelines for the community and advocate the brand on Social Networks so that the community can have the potential customers well connected.

Rest the task or work done by both the managers like removing, banning users, their comments or posts etc are very basic and also pointed in other answers.

Community manager is a person who publicly supports or recommends a cause or policy on behalf of a company to the company members. He also gives feedback of consumers to the company.

Roles of Community manager

1. Content creation

2. Add or remove new members from the community

3. Public relations

4. Customer relations

5. Provide support to the all the members

6. Introduce community members to each other

7. Create online / offline contest or event
    

Roles of Social media manager

1. Product or service development

2. Identify target customers

3. Content strategy

4. Engagement strategy

5. Promotion strategy

6. Analyze and measure (Growth, Reach, Leads, Quality)

 

Community Manager is the face of the company, One who is a bridge between the Organization and the Legal or Corporate or Media People. He/ She is reasonably responsible for two way communication of the company inwards and outwards. Community Manager has to be Digital Savvy, he.she has to have a knowledge and responsibilities of WEB 2.0 professional. He/She has to incorporate online tools of automation as well as create inter-personal network for the company. He/She has to actively participate in both online and offline networking of the company. Community manager takes care of the Content Creation, Public Relations, Social Media Marketing, Event and Event Planing, Customer Relation, Communication/Marketing Strategy, Data Analysis through Google analytics and Last but important All round business development of the organization.

Social Media Manager on the other hand will only administer social media profile and handle it with pure finesse. He/She has to deliberate plan and set goals for the development of the brand made by the company or the company itself. He/She has to create brand awareness and maintain online reputation. He/she has to keep eye on the competitor's social profile to determine the factors to develop and implement strategies helpful for the company. Social Media Manager has to look after content marketing, content creation for the social channels( blogging, micro blogging, blog snippets), Search Engine Optimisation, Generating Inbound traffic, Pushing sales by promoting brand with affiliate marketing, Target potential customers, Visual design and Web development through photoshop skills. Social media manager main goal is to maintain, measure and analyze  the  ROI and KPI of the brand and company. Promoting content to have potential customers to convert into sales which generates lead in the company.

Social Media is one of the most important Channel to promote the business direct to the cutsomers,where as Community Manager Deals with the people who never heard about the business and boost the business perception of the company.

Difference :

Social Media manager is more focus on the Branding of the product on the social channels by managing the post creations, micro blogging, schedule posts, and shared to various other platforms. Social Media comes first in every Brand to promote.

A community manager is the overall person in the community that handles both the outreach and management of the social media channels, moderation, press release and content strategy.

Social media manager and Community manager sound same but their functions and responsibilities are actually quite different.

 

The “Social Media Managers’”main goal is to approach the growth, their key responsibilities are:

 

1. He creates a brand on all social media platforms.

2. Valuable content is the most important aspect to drive the traffic, so they try to generate content to drive more and more traffic

3. Engaging customers with the product is their primary focus.

4. Focus on planning and making strategy for various brand channels

 

Whereas, a Community Manager is often seen as someone who keeps uplifting the people around him. He is the face of the brand in the digital arena and act as an adviser for his customers.

 

1. They are responsible for building and protecting the relationships with key members and facilitating customers engagement.

2. He collects various information internally from blogs, forums, and various social media platforms.

3. They find the way to reach the right people to target and engage with.

4. They spread the awareness about the company who do not know about them.