'Get started' usually is used in relation to some individual task or project. It emphasizes the initial work that has to be done to start something in the first place - organize a work chart, decide what the goals are, or simply to stop what we are doing and start work on the project or task. It usually refers to some task that will take some time. 'Start' is used more if there is some general change in attitude or behavior. "Let's start being nicer to one another." "Let's start being more careful when driving." You can actually use either one, but the situations where they are usually used are different.

"Let's get started" has a more informal, friendly tone than using the verb "to start" on its own. This type of construction is not used in formal writing, but is restricted to spoken language or informal writing.