Hermite and Laguerre $\beta$ ensembles are two of the most well studied models in random matrix theory with the special cases $\beta=1,2,4$ corresponding to eigenvalue distributions of classical Gaussian and Wishart ensembles. Using tridiagonal matrix models for these ensembles introduced by Dumitriu and Edelman, Ramirez, Rider and Virag established Tracy-Widom $\beta$ scaling limits for the largest eigenvalues. They also showed that the upper tail decays of the Tracy-Widom $\beta$ distribution decays like $\exp(-2\beta t^{3/2}/3)$ while the lower tail decays like $\exp(-\beta t^{3}/24)$.  I shall describe some recent work establishing sharp tail estimates in the pre-limiting models up to $1+o(1)$ error at the level of exponents, improving earlier results by of Ledoux and Rider. I shall also describe a number of applications of these estimates to exactly solvable planar last passage percolation, which are related to the classical $\beta$ ensembles via some remarkable distributional identities.