Basically, you can modify the DP for Levenshtein edit distance to compute distances for your problem. The Levenshtein DP amounts to finding shortest paths in an acyclic directed graph that looks like this
*-*-*-*-*
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where the arcs are oriented left-to-right and top-to-bottom. The DAG has rows numbered 0 to m and columns numbered 0 to n, where m is the length of the first sequence, and n is the length of the second. Lists of instructions for changing the first sequence into the second correspond one-to-one (cost and all) to paths from the upper left to the lower right. The arc from (i, j) to (i + 1, j) corresponds to the instruction of deleting the ith element from the first sequence. The arc from (i, j) to (i, j + 1) corresponds to the instruction of adding the jth element from the second sequence. The arc from (i, j) corresponds to modifying the ith element of the first sequence to become the jth element of the second sequence.
All you have to do to get a quadratic-time algorithm for your problem is to define the cost of (i) adding a datapoint (ii) deleting a datapoint (iii) modifying a datapoint to become another datapoint and then compute shortest paths on the DAG in one of the ways described by Wikipedia.
(As an aside, this algorithm assumes that it is never profitable to make modifications that "cross over" one another. Under a fairly mild assumption about the modification costs, this assumption is superfluous. If you're interested in more details, see this answer of mine: Approximate matching of two lists of events (with duration) .)