Channel-aware scheduling strategies have emerged as an effective mechanism for improving the throughput of wireless data users by exploiting rate variations. The improvement in throughput comes however at the expense of an increase in the variability of the service rate received over time. While the larger variability only has a limited impact on delay-tolerant data transfers, it does severely affect delay-sensitive applications. In order to examine the merits of channel-aware scheduling for the latter users, we consider a wireless system supporting a combination of streaming and elastic traffic. We first examine a scenario with rate-adaptive streaming traffic, and analyze the flow-level performance in terms of transfer delays and user throughputs for various canonical resource sharing schemes. Simulation experiments demonstrate that the analytical results yield remarkably accurate estimates, and indicate that channel-aware scheduling achieves significant performance gains. Next we investigate a scenario where the streaming sources have an intrinsic rate profile and stringent delay requirements. In that case, channel-aware scheduling yields only modest performance gains, and may even be harmful.