This study is about degenerate Hermite Appell polynomials in three variables or ∆ h-Hermite Appell polynomials which include both discrete and degenerate cases. After we recall the definition of these polynomials and special cases, we investigate some properties of them such as recurrence relation, lowering operators (LO), raising operators (RO), difference equation (DE), integro-difference equation (IDE) and partial difference equation (PDE). We also obtain the explicit expression in terms of the Stirling numbers of the first kind. Moreover, we introduce 3D-∆ h-Hermite λ-Charlier polynomials, 3D-∆ h-Hermite degenerate Apostol-Bernoulli polynomials, 3D-∆ h-Hermite degenerate Apostol-Euler polynomials and 3D-∆ h-Hermite λ-Boole polynomials as special cases of ∆ h-Hermite Appell polynomials. Furthermore, we derive the explicit representation, determinantal form, recurrence relation, LO, RO and DE for these special cases. Finally, we introduce new approximating operators based on h-Hermite polynomials in three variables and examine the weighted Korovkin theorem. The error of approximation is also calculated in terms of the modulus of continuity and Peetre's K-functional.