There is no significant difference between “something being on fire” and “something burning”, where “something” is a flammable material. However, the two expressions are not interchangeable in many other situations. For example, one can say that “a fire is burning”, but not that “a fire is on fire”:
Another difference is that “burn” can be used transitively (i.e. “someone can burn something”), but “being on fire” cannot:
(Note that “burned” is the preferred form in the US and “burnt” in the UK.)
The transitive form of “to be on fire” is “to set on fire”, which is synonymous with “to ignite, to cause to begin to burn”, but that is not the same as “to burn”. When something was burnt (or burned), the implied meaning is that it was destroyed, whereas when it was merely set on fire, it was in flames, but it may still be undamaged. You could say, for example,
Other differences between “burning” and “being on fire” are related to their idiomatic meanings. “On fire” can be used in the sense of “achieving great results”, used especially in sports commentary, e.g.
“Burning”, on the other hand, can be used in the sense of “urgent and important”, usually when referring to a question or an issue, e.g.
or in the sense of “strong, extreme” when talking about feelings, e.g.