Why is there an additional capacitor needed in DRAM cell then?
Because a memory cell needs at least 1 switch and 1 memory element for it to be addressable, a minimum of two elements. There are memories like SRAM can use only transistors in a bi-stable configuration. From wikipedia "Memory cells that use fewer than four transistors are possible – but, such 3T or 1T cells are DRAM, not SRAM."
Wouldn't be drain/source to bulk capacitance providing the same
functionality if there was no capacitor?
A capacitor is still a capacitor even if it is just between the drain and source. Yes, it could provide the same functionality, but why would you add and dope extra silicon and take up more space if you could just create a capacitor with p material? That's right, go for the capacitor to save costs.
Guess which one is smaller:
If it is the case, is the capacitor connected to the node for
specifically larger capacitance?
Yes, the capacitor is connected to the transistor for larger capacitance. The industry is trying to make cells smaller all of the time so they have to invent new geometries to store enough electrons that the voltage doesn't disappear the next time the cell is read.
Remember, capacitors and transistors are nice ways for us to model what is going on with billions of electrons and different materials.