I agree with Emerton's answer (and had similar thoughts, but since he is a leading expert in this field, it's better if the answer comes from him). I would say that if anything the opposite is true: nowadays people (especially those of an arithmetic bent) are more interested in $p$-adic cohomology than $\ell$-adic cohomology and the former is viewed as richer and more difficult than the latter. Thus the importance of Dwork's work could scarcely be higher.
Flipping to the other side of the Weil conjectures, I think it is also not quite fair to say that Deligne proved the Riemann hypothesis "with Grothendieck's methods". I know you mean that he used Grothendieck's methods ($\ell$-adic cohomology) rather than Dwork's methods ($p$-adic analysis) but it doesn't do justice to the range of new ideas that Deligne brought to the table (as well as the ideas that were left on Grothendieck's table, to Grothendieck's eternal consternation).
Flipping back again, note that the Weil conjectures (and even parts of Deligne's significant generalization "Weil II") have since been proven completely by $p$-adic methods, c.f. the work of Kiran c.f.:
Kedlaya, Kiran S.(1-MIT) Fourier transforms and $p$-adic `Weil II'. (English summary) Compos. Math. 142 (2006), no. 6, 1426--1450.
This is another sign that the "Dwork school" is going strong in contemporary research.
