People are believing things that are not true. と People believe things that are not true. はどう違いますか？説明が難しい場合は、例文を教えて下さい。

The same difference as between the simple present tense and the present continuous tense in other situations.

#2 uses the simple present tense and so it sounds more like a general statement about the nature of mankind / of people, about what is normal for people.

#1 uses the present continuous tense and so it sounds like a more focused statement - it sounds like it is talking about a smaller time frame. For example, maybe recently the newspapers have been printing lies or inaccurate information and so people are believing things that are not true these days. Maybe normally the newspapers are more accurate and trustworthy and so this is not the normal situation for people.

If we really want to then we can use these tenses to talk about other time frames (e.g. "people have been believing things that are not true for 12,000 years" - a very long time) but what I explained above is how the tenses 'feel' if they don't have any extra information.