Accepted answer
The single most important distinction is between what purity measures — the fraction of detected material that is your target — and what you actually want to know — how much of the material in the vial is your target.
Temperature affects the dynamics of molecular conformation, and if a peptide has proline residues that interconvert on the chromatographic timescale, the peak will split or shoulder at low temperature and collapse at high temperature.
Buffer versus acid in the mobile phase changes the ionisation state of basic and acidic residues, shifting retention and selectivity — same vial, potentially different separation.
The ICH Q3A impurity thresholds and the relevant pharmacopoeial chapters all specify method validation requirements that almost no research-grade certificate claims to meet.
Compare purity within a single laboratory on the same method, never across laboratories.