A comparative study was performed in this work to investigate the crystal structure, thermal behavior, melting behavior, overall isothermal melt crystallization kinetics, and spherulitic morphology of biodegradable neat poly(ethylene adipate) (PEA) and its novel copolyester poly(ethylene adipate-co-5 mol % ethylene succinate) P(EA-co-5 mol % ES). Both neat PEA and P(EA-co-5 mol % ES) show the same crystal structure. Relative to neat PEA, the incorporation of a small amount of ethylene succinate (ES) composition exerts almost no influence on the glass-transition temperature, reduces the melting point temperature and equilibrium melting point temperature slightly, but suppresses the nonisothermal melt and cold crystallization significantly of P(EA-co-5 mol % ES). Depending on crystallization temperature, both neat PEA and P(EA-co-5 mol % ES) show double melting endotherms or one melting endotherm, which can be well explained by the melting, recrystallization, and remelting mechanism. The overall isothermal melt crystallization kinetics was studied for neat PEA and P(EA-co-5 mol % ES) and analyzed by the Avrami equation. With increasing crystallization temperature, the crystallization rates are reduced for both neat PEA and P(EA-co-5 mol % ES); moreover, the crystallization rate is faster in neat PEA than in P(EA-co-5 mol % ES) at a given crystallization temperature. The crystallization mechanism remains unchanged despite crystallization temperature and the incorporation of ES composition. Depending on crystallization temperature, neat PEA and P(EA-co-5 mol % ES) may form ring-banded or ringless spherulites.