So my teacher said that we can use both to describe a planned plan or idea. Is there any difference between the two? For examples: "The party will be starting at 7" vs "The party is going to start at 7".
They are very similar. Your examples are basically the same, and so the teacher is correct.
The future continuous refers to an unfinished action or event that will be in progress at a time later than now.1
The use of going to refer to future events suggests a very strong association with the present. The time is not important, it is later than now, but the attitude is that the event depends on something in the present situation that we know about. Going is mainly used to refer to our plans and intentions or to make predictions based on present evidence. In everyday speech, going to is often shortened to gonna, especially in American English, but it is never written that way.2
Various future tenses include:3
- Simple future tense
- Future continuous tense
- Future perfect tense
- Future perfect continuous tense