This is a more complicated subject when comparing Caffe vs TensorFlow. Both have fragmented into multiple branches and missions, making direct speed comparisons increasingly difficult. Even in the rare instances where objective and/or recent benchmarking is available, the most meaningful use cases are hidden behind the proprietary internal politics of Google and Facebook. After all, they are the highest-volume creators and consumers of both frameworks, respectively.
In the case of TensorFlow, one has to evaluate both the data center/desktop/VM environment and the recently-launched Lite iteration, which is arguably more comparable to the mobile-centric mission of many aspects of Facebook's Caffe2.
Over the last five years, TensorFlow has been popularly estimated to lag behind even the original version of Caffe, since the latter has been perceived to benefit from its low-level streamlined approach to neural network design. However, this judgement doesn't account for the improved performance of TensorFlow 2, the impact of which is likely to become clearer over the next 18 months.
Neither does it account for the speed advantages that scale and uptake bring to the table: when a framework reaches a certain level of diffusion, ubiquity tends to compensate for architectural bloat via dedicated industry workarounds, which come into existence through an increased market demand. These include ASIC solutions (in this case, via Google's own TPUs), edge caching and other case-specific resources available through low-latency local data centers and bespoke on-premises hardware solutions.
Like-for-like speed testing between TensorFlow and Caffe is a problem at the moment, due to increased recent activity in their release cycles, the difference in scope between various versions of both frameworks, and the fact that Caffe is still primarily used for vision-related tasks—which is an important but not pivotal element in TensorFlow.
Iflexion recommends: If your project is centered around vision-based research, and particularly if it incorporates facial recognition, it's a reasonable bet that a PyTorch/Caffe2 basis will provide a more performant solution. Facebook's vanguard position in facial recognition technologies, facilitated by its unrivalled access to high-volume social network facial image data, has inclined it toward Caffe for a reason. You'll also benefit from an improved API and more configurability and flexibility than the original Caffe.
For all other purposes, TensorFlow's scalability, pace of innovation and industry support suggests a likely edge in production-level performance across all phases of development and deployment.