Short answer
If the empty cell or tray says CR2032, do not buy CR1632 because both are “3.2 mm lithium coins.”
DL1632 / ECR1632 map to CR1632; DL2032 / ECR2032 map to CR2032 — still different diameters.
What is the same
Both are 3-volt primary lithium manganese dioxide coin cells about 3.2 mm thick. The “32” in the code is the thickness cue in millimetres × 10.
What is different
Diameter: CR1632 ≈ 16 mm; CR2032 ≈ 20 mm. A CR1632 will rattle or lose contact in a CR2032 tray; a CR2032 will not fit a CR1632 compartment.
Capacity and common uses differ with size — CR2032 is the mass-market tracker / CMOS / key-fob cell; CR1632 appears in thinner or smaller trays that name it explicitly.
Do not use list
Do not substitute CR1632 for CR2032 (or the reverse) because both end in 32.
Do not confuse either cell with CR1620 (same ~16 mm class, thinner) or CR2025 (same ~20 mm class, thinner).