Endorsement is usually more official, or it is done by an important person or organization. Often, it can be used for political candidates. A regular person would support a candidate, but a famous, important person or organization would endorse a candidate. It's in a more official capacity. 'To endorse' something can also mean to sign your signature on a legal document...here's the official/important/binding aspect again. 'To support' is most often used generally, i.e. to support a cause, to support a person, to support a decision, etc.