For HK mobiles the mobile owner always has to pay "airtime" for both incoming and outgoing calls. For outgoing international calls there is then an additional charge depending on the country. The Peoples deal is that your airtime is HK$0.25/min and on that deal they will not charge you anything additional for international calls to the 23 listed destinations. But not that these 23 destinations do not include countries where the calling party pays to call mobiles (e.g. UK, Australia). For calls to Australia mobile there would be a significant additional charge from Peoples.
0060 is an international calling service that can be used from any phone as an alternative to that phone provider's international charge. Obviously if you have this Peoples package you wouldn't want to use 0060 to call any of the 23 destinations because the international charge from Peoples is zero. However, to call, say, Australia mobile from a Peoples phone you might find that it is cheaper to use 0060 than Peoples own IDD charges. This would depend on the particular deals you are able to get from Peoples or 0060 (PCCW).
There are many other postpaid international calling solutions available as well as 0060, including 0080 (HGC), 1529 (IDT), Zone1511, etc.
You would never use an international calling solution for local calls - these are always free (to both parties) for landline-to-landline, and always incur the airtime charge to the mobile subscriber for each end that is mobile.
Note that many mobile subscription (postpaid) packages include large bundles of airtime within the monthly subscription. It can be a very complex calculation to work out what the best combination of deals will be absolutely the best for your circumstances. And unless you calling pattern is very predictable then you may never be able to always pick the best deal in advance.