Why does my fluticasone nasal spray say on the leaflet "use caution when taking with prednisone"? My doctor prescribed me both for bronchitis.

Precaution. The fda mandates certain wording when it comes to steroids. Fluticasone is a steroid but when used as directed should not pose through the body side effects. Prdenisone pills could cause side effects and should be used only when other medicnes are not effective.
Steroid. The reason for the caution is both products contain steroids. The nasal steroids don't likely get absorbed the systemic circulation meaning the whole body. You should discuss with your doctor or possibly stop the nasal steroids while taking the oral steroids as taking both may be redundant.