The ability to produce timely and accurate flu forecasts in the United States can significantly impact public health. Augmenting forecasts with internet data has shown promise for improving forecast accuracy and timeliness in controlled settings, but results in practice are less convincing, as models augmented with internet data have not consistently outperformed models without internet data. In this paper, we perform a controlled experiment, taking into account data backfill, to improve clarity on the benefits and limitations of augmenting an already good flu forecasting model with internet-based nowcasts. Our results show that a good flu forecasting model can benefit from the augmentation of internet-based nowcasts in practice for all considered public health-relevant forecasting targets. The degree of forecast improvement due to nowcasting, however, is uneven across forecasting targets, with short-term forecasting targets seeing the largest improvements and seasonal targets such as the peak timing and intensity seeing relatively marginal improvements. The uneven forecasting improvements across targets hold even when "perfect" nowcasts are used. These findings suggest that further improvements to flu forecasting, particularly seasonal targets, will need to derive from other, non-nowcasting approaches.