The short answer
In everyday conversation, people use "compliance check" and "investigation" to mean the same thing, and that's not unreasonable, because both describe HMRC looking into your tax affairs. But HMRC's own literature reserves the word "investigation" more for the serious end of the spectrum, particularly cases run under Code of Practice 8 or Code of Practice 9.
A routine compliance check, such as a query about one expense claim, is not the same scale of matter as a Code of Practice 9 case, where HMRC suspects deliberate fraud and the taxpayer may need to make a formal disclosure to avoid criminal prosecution. Both fall under the broad HMRC compliance umbrella, but the stakes and process differ enormously.
The practical point is not to panic simply because you've heard the word "investigation" attached to your situation. Check exactly what type of letter you have and whether it references a specific Code of Practice.

