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Husk, (削除) 10 (削除ここまで) 9 bytes

Lṁ-→FnmṖ1

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Inspired by ais523's Brachylog solution, finds the longest common (not necessarily contiguous) subsequence and subtracts it from each string.

(削除) I don't like having to repeat Ṗ twice and having to explicitly refer to the two arguments 1 and 3, but I couldn't find any shorter alternative. (削除ここまで) Now I'm happy :)

Explanation

Lṁ-→FnmṖ1 Input: a list containing the two strings
 Ṗ Get all possible subsequences
 m 1 for each string
 Fn Keep only the common ones
 → Get the last one (which will be the longest)
 - and subtract it
 ṁ from each of the inputs, then concatenate the remaining characters
L Find the length of this

Subtracting the longest common substring for each input results in the lists of characters that need to be deleted/inserted. We concatenate these two lists and get the length to calculate the edit distance.