If I use StreamStats at an existing gage location, will I get the same results as if I computed statistics using the gage records?

If you use the Query Streamgage tool to get the statistics, then the statistics should be the same as what you computed if the same period of record was used, and if the statistics are interpretive (peak- and low-flow frequency statistics), the same interpretation of the data was made.  If you obtained estimates using the the Regression-Based Scenarios tool, then the estimates will be different, as they will be computed from regression equations rather than from the actual record at the streamgage.  The regression equations provide estimates that represent somewhat of an area-based average of the peak-flow statistics for the streamgages in the region in which your site of interest is located. Flows at the streamgages used to develop the equations generally are minimally affected by human alterations. As a result, the equations provide estimates of flows that would be expected under essentially natural flow conditions.