Separations between testing and tolerant testing for (natural) classes of functions?
The recent preprint of A. Levi and E. Waingarten [1] establishes such a separation* for two "(arguably) natural" properties of functions: unateness and junta-ness.

*(for junta-ness, such a separation is shown only for non-adaptive testers.)


  Theorem 1. ([1]) Non-adaptive tolerant testing of $k$-junta-ness has query complexity $\tilde{\Omega}(k^2)$.


On the other hand, Blais [2] showed that non-adaptive testing of $k$-juntas had query complexity $\tilde{O}(k^{3/2})$, hence the separation.


  Theorem 2. ([1]) Non-adaptive tolerant testing of unateness has query complexity $\tilde{\Omega}(n^{3/2})$.


On the other hand, Baleshzar et al. [3] showed that non-adaptive testing of unateness had query complexity $\tilde{O}(n)$, hence the separation.


  Theorem 3. ([1]) Tolerant testing of unateness has query complexity $\tilde{\Omega}(n)$.


On the other hand, Chen, Waingarten, and Xie [4] showed that non-adaptive testing of unateness had query complexity $\tilde{O}(n^{3/4})$, hence the separation.



[1] Erik Waingarten and Amit Levi. Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs, 2018. arXiv:1805.01074

[2] Eric Blais. Improved Bounds for Testing Juntas. APPROX-RANDOM, 2008.

[3] Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, C. Seshadhri. Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps. ICALP, 2017.

[4] Xi Chen, Erik Waingarten, Jinyu Xie. Boolean Unateness Testing with $\tilde{O}(n^{3/4})$ Adaptive Queries. FOCS, 2017.