The BeiDou Navigation Satellite System (BDS) has completed third phase construction and currently provides global services, with a mixed constellation of BDS-2 and BDS-3. The newly launched BDS-3 satellites are equipped with rubidium and passive hydrogen maser (PHM) atomic clocks. The performance of atomic clocks is one of the cores of satellite navigation system, which will affect the performance of positioning, navigation and timing (PNT). In this paper, we systematically analyze the characteristics of BDS-2 and BDS-3 atomic clocks, based on more than one year of precise satellite clock products and broadcast ephemeris. Firstly, the results of overlapping Allan variations demonstrate that BDS-3 Rb and PHM clocks improve better in stability than BDS-2 Rb clock and are comparable to GPS IIF Rb and Galileo PHM clocks. Accordingly, the STDs of BDS-3 broadcast satellite clock are better than GPS and BDS-2, which are at the same level with that of Galileo. Secondly, the inter-system bias (ISB) between BDS-2 and BDS-3 is analyzed by satellite clock datum comparison and precise point positioning (PPP). Surprisingly, the discrepancy between BDS-2 and BDS-3 satellite clock datum has a great difference between products that could reach up to about 10 ns for WHU satellite clock products and broadcast ephemeris. Moreover, the ISBs between BDS-2 and BDS-3 satellite clocks are quite stable over one-year periods. Thirdly, due to the improved stability of BDS-3 atomic clock, the 68% positioning accuracy is better than 0.65 m at 10 min for BDS-3 PPP, based on broadcast ephemeris. Besides, the non-negligible bias between BDS-2 and BDS-3 will greatly affect the BDS precise data processing. The accuracy of positioning is greatly improved when considering the ISB.