To establish a serial connection between the PICAXE and a PC, resistors are not actually required, however it is recommended in order to protect the systems should a short circuit or other erroneous event occur. The same also applies to a PIC. If you are referring to the two download resistors used for the PICAXE, (using the values recommended in the documentation) the 10k resistor keeps the serial in pin low while the 22k resistor limits current into the pin (which may help the internal clamping diodes to work).
Using such an approach, the PIC and PC can communicate serially, however by adding a RS-232 transceiver chip such as a MAX232, it is possible to get better performance. To elaborate, the MAX232 chip does (most simply) two things: inverts the bit stream and amplifies the drive voltage. It is this which makes a normal bit stream of 1s and 0s "RS-232 compatible". Typically we refer to low=0=0V and high=1=5V, however RS-232 uses negative logic with low=+ve voltage and high=negative voltage. The extra buffering may allow for longer transmission distances and/or better reliability in noisy environments at higher bit rates. RS-232 transceiver chips also often advertise high ESD protection (which can be an issue, especially with transmission lines which may not always be always conencted at the other end), with the MAX232 advertised to be around 15kV or so...
With regard to signalling voltages, trusty Wikipedia says:
...signal levels of ±5 V,±10 V,±12 V, and ±15 V are all commonly seen...
And further to talk about voltage protections:
RS-232 drivers and receivers must be able to withstand indefinite short circuit to ground or to any voltage level up to +/-25 volts...
A PICAXE (or PICmicro) will probably curl its toes and run away if you tried subjecting it to these conditions!
So long-story-short, there is nothing special about how the PICAXE bootloader implements the serial communications systems - technically whatever the PICAXE can do, a plain PICmicro can do as well. However the system typically used to connect the PICAXE to a PC is very simple because that's the way we like to do things!!! Keep it simple! However, by all means add a RS-232 chip if you want, but remember that the signal levels are inverted and you'll need to compensate for that on either the receiver or transmitter end (use T4800 instead of N4800 when writing your PICAXE code, for example).
ylp88