If you include mathematical physics (string theory, QFT) to "applied mathematics", the breakthroughs are too numerous to list them here.
Here is a very recent example not related to QFT or string theory. An important Connes Embedding Conjecture from the theory of von Neumann algebras, has been apparently disproved by the arguments from Computer Science (quantum computing):
(The complete paper is not published yet, given its volume, it will take some time to verity).
More examples (just a selection of those which I encountered during my career).
In 1987, Zograf and Takhtadzhyan made a breakthrough in the classical accessory parameter problem of uniformization theory by proving a conjecture of physicists Belavin, Polyakov and Zamolodchikov inspired by string theory.
In 1993, Kari Astala proved the old conjecture of Gehring and Reich by using ``thermodynamic formalism''. The proof has been substantially simplified since, but still uses the main idea which comes from statistical physics. In 2020 a a surprising connection of this former Shapiro conjecture with SLE has been found.
In the early 2000s Oded Schramm created the SLE theory (Stochastic Loewner Evolution, aka Schramm-Loewner evolution) on the boundary of probability and complex analysis. It was directly inspired by questions from physics (statistical mechanics, percolation theory).
In 2005, Mukhin, Tarasov and Varchenko proved the B. and M. Shapiro Conjecture from real algebraic geometry by unexpected argument from mathematical physics (Bethe-ansatz from statistical mechanics).