The best way to differnciate is to look at the count-rate vs time graph. Random nature of radioactivity is indicated by the fluctuation of the count-rate. Whereas, spontaneous means that the process of radioactivity is not dependent upon any external environmental conditions like temperature, pressure, magnetic/electric field. For example: Plot a count rate graph at 100 degrees and then at 200 degrees. Graph of the shape will be the same and also half life in both the cases will be same indicating the spontaneous nature of the process of radioactivity.

The best way to differentiate is to look at the count-rate vs time graph. Random nature of radioactivity is indicated by the fluctuation of the count-rate. Whereas, spontaneous means that the process of radioactivity is not dependent upon any external environmental conditions like temperature, pressure, magnetic/electric field. For example: Plot a count rate graph at 100 degrees and then at 200 degrees. Graph of the shape will be the same and also half life in both the cases will be same indicating the spontaneous nature of the process of radioactivity.