In this paper we introduce the differential meet-in-the-middle framework, a new cryptanalysis technique for symmetric primitives. Our new cryptanalysis method combines techniques from both meet-in-the-middle and differential cryptanalysis. As such, the introduced technique can be seen as a way of extending meet-in-the-middle attacks and their variants but also as a new way to perform the key recovery part in differential attacks. We apply our approach to SKINNY-128-384 in the single-key model and to AES-256 in the related-key model. Our attack on SKINNY-128-384 permits to break 25 out of the 56 rounds of this variant and improves by two rounds the previous best known attacks. For AES-256 we attack 12 rounds by considering two related keys, thus outperforming the previous best related-key attack on AES-256 with only two related keys by 2 rounds.