What is the difference between teaching content and teaching skills in language learning?

Great question!

The language skills are reading, writing, listening, speaking (in no specific order). Teaching those skills may include some instruction, but the class and homework activities have to do with practicing those skills. For example, for speaking, students have conversations, interviews, answer oral questions about a reading, join class discussions, present projects, etc. All of those activities are based on, or involve speaking.

Content can include grammar rules, vocabulary lists, and cultural information, to name a few. So the lesson on how to conjugate verbs in the future tense is content. How people in the country (where the language is spoken) celebrate holidays, or what they eat for breakfast, or what a map of the country looks like are all content.