Understanding LiPo charging / protection circuit
Protection circuits are usually distinct from charging circuits. Many battery packs are designed with the intention of being charged by a dedicated unit that will control the charging process. 

The charging process may involve cell balancing, if the pack contains a large number of cells in series, generally 4+ cells in series (4S, 14.4V) nominal will require balancing, 3S and lower it's also a good idea to balance for the health and longevity of your battery, but not necessarily required. Balancing circuits can get complicated and typically involve a BMS (battery management system) composed of a dedicated IC and multiple external mosfets, There's a project on github that seeks to create an openBMS system. This might be a good resource if you are looking for more information. 

The charging cycle for lithium ion batteries can be quite complex, especially in the case of multiple cells in series, but typically involves 4 basic steps: 


Read voltage, if lower than a certain value (typically 2.8V or so for Li based cells) then begin trickle charge until cell reaches safe charging level, doing this avoids damaging the cell.
constant current charging: the cell is charged at a constant current, typically .5C-1C for normal charging, e.g. for 1000mAh battery, charge at between 500mA-1000mA.
constant voltage charging: once battery reaches a certain point (typically about 60% of total charge (3.8V or so) begin charging at the target final voltage (4.2V for normal 1000 charge cycle expected life) you can go higher, and it will give you more battery life, but it will reduce the life of the battery.
Maintenance charging: The battery has a natural discharge rate in the neighborhood of 8% at 21 °C per month, when the battery drops below 10% of full charge, recharge to target voltage using constant voltage charging. This is configurable depending on application.


Notes: there are ICs that will handle the bulk majority of this for you, otherwise you will need to resort to designing an MCU controlled circuit with external boost/buck converter, or linear regulator.

The protection circuit (PCM) is fairly simple and often times integrated right into the individual cells, these cells are typically labeled: protected, or unprotected. The PCM will monitor things like: input voltage, output current, cell voltage, temperature, etc. They often times are not that robust and should be considered a last resort in a critical system. Alarms should be going off if the PCM is ever triggered. 

To answer your specific question: the DW01 is optimized to favor the charger in the case of overcharge, so the charger would remain connected to the circuit, suppling nessesary voltage, while the battery is disconnected, TP4046 looks like it is designed to handle up to 8V and as a linear charger it will be dissipating excess voltage as heat. These ICs are often thermally protected and will automatically shut off in the case of excess current or voltage. If there is an overcharge, it means the circuit is likely destroyed, thus it is nice to try to disconnect the battery in this case, as it would would pose a significant safety risk.

Also, remember that mosfets contain internal diodes, so even if charge protection mosfet were to be turned off, the battery would still be attached to the circuit as long as the voltage on the drain side of the mosfet is below a certain voltage.