Simply,
Specificity - method separates and measures the analyte in the presence of junk.
stability-indicating - methods detect analyte degradation when exposed to various stresses, such as heat, light, acid, oxygen etc. Often the HPLC method separates and detects degradation products of the analyte.
Refer to ICH for more detail...
From ICH Q2 Revision 1
VALIDATION OF ANALYTICAL PROCEDURES: TEXT AND METHODOLOGY
ICH Harmonised Tripartite Guideline
2. SPECIFICITY
Specificity is the ability to assess unequivocally the analyte in the presence of components which may be expected to be present.
Typically these might include impurities, degradants, matrix, etc.
Lack of specificity of an individual analytical procedure may be compensated by other supporting analytical procedure(s).
This definition has the following implications:
Identification: to ensure the identity of an analyte.
Purity Tests: to ensure that all the analytical procedures performed allow an accurate statement of the content of impurities of an analyte, i.e. related substances test, heavy metals, residual solvents content, etc.
Assay (content or potency): to provide an exact result which allows an accurate statement on the content or potency of the analyte in a sample.
For stability-indicating, there may be no single assay, but the description is often applied to some HPLC methods that have been validated for specific drug substances or products to detect degradation products under specified stress test conditions ( refer ICH Q1 ). There is some information in ICH Q5C for Biotech products, but there may be similar/better information in other ICH Q series documents - I didn't check.
5. STABILITY-INDICATING PROFILE
On the whole, there is no single stability-indicating assay or parameter that profiles the stability characteristics of a biotechnological/biological product. Consequently, the manufacturer should propose a stability-indicating profile that provides assurance that changes in the identity, purity and potency of the product will be detected.
At the time of submission, applicants should have validated the methods that comprise the stability-indicating profile and the data should be available for review. The determination of which tests should be included will be product-specific.