Maxim is correct that ASN/AS16 only report "named" storms with winds greater than or equal to 60 knots.
If when flying over a known storm area, the storm is not available in the quick search hurricane tool, you will be shown only what local stations are reporting as cloud and wind activity. If the metar reports for the area and nearby have extreme data associated with them suggesting an active convergence, ASN/AS16 will generate TS and most certainly rain.
When a named storm is available and you have "Depict Hurricanes" enabled in the options, the main map will display a generic hurricane cloud pattern over the area and the graphic along with its associated extreme weather conditions will move over time based on real world geo data. You may have to zoom out to see the symbol, otherwise the map will display the precip patterns in the area. In the sim, FSX/P3D is not capable for displaying circular cloud patterns as you would find in a tornado or hurricane eye. Instead, you will see a large area of tightly grouped and intense thunderstorms over a wide area in a roughly circular pattern.
It is not unusual for some data sources and ASN/AS16 data sources to disagree as to when a tropical storm gains or loses hurricane status. Storms within 100 miles of land regardless of their status will have very active display in the sim since the metars in the area will be reporting rain, wind and TS. Over remote areas of the ocean far away from actively reporting stations, if AS is not listing a named storm in the intro page of the UI or in the search, the area in the sim may not look anything special or intense unless a real world aircraft flies there and provides a sigmet/airmet. One way to get an idea if an area where a non-named storm may have extreme conditions is to focus on that area of the map with sigmet/airmet layer on and look for a red pentagram. When Hermine hit western Florida recently, it was not being reported as a named storm in AS16, but when I flew there, I had intense TS activity as a result of the metars in the area and my aircraft broke up.
-Pv-