This is (almost) a question in financial engineering. First I will note that a discussion of "the greeks" is well presented at https://en.wikipedia.org/wiki/Greeks_(finance) These measures are first, second and higher order derivatives (or rate of change comparisons) for information that is generally instantaneous. (Bear with me.) For example the most popular, Delta, compares prices of an option or other derived asset to the underlying asset price. The reason we are able to do all this cool analysis is because the the value of the underlying and derived assets have a direct, instantaneous relationship on each other.
Because beta is calculated over a large period of time, and because each time slice covered contributes equally to the aggregate, then the "difference in Beta" would really just be showing two pieces of information:
- Impact of today's asset and market price on Beta
- Impact of (today minus 3 years)'s asset and market price on Beta
Summarizing those two pieces of information into "delta beta" would not be useful to me.
For further discussion, please see http://www.gummy-stuff.org/beta.htm specifically look at the huge difference in calculation of GE's beta using end-of-month returns versus calculation using day-before-end-of-month returns.