An intriguing question. In some cases - yes, in some no.
Specs - eg 28X2
4 x program slots with up to 3200 lines of program code in each and if that's not enough you can put program code in external memory too! Up to 64MHz clock speed, 23 I/O pins that can be used as inputs or outputs in any combination, 9 ADC inputs. Supports Serial (RS232), I2C, SPI, UNI/O, and more protocols.
Commodore 64. Slower clock speed. But compare Basic on both, and they might be both roughly around 1000 instructions a second. It depends on interpreted vs compiled basic.
Flexibility to run other languages eg C. Picaxe will lose on that one.
Memory - Commodore 64 is 64k of memory. That is quite a lot more than a picaxe. 64000 lines of assembly code vs 3200 lines of basic, but then again, one line of basic often needs a number of lines of assembly, even with an efficient compiler. Try "DIM MYARRAY(10000)" and see how much memory you have left.
I2C and SPI are not available on a vintage computer, so the picaxe wins there.
Keyboard with buffer - I think the commodore wins on that one but maybe one of those cores on a picaxe can be devoted to running a keyboard.
Display. I think the commodore wins there as the picaxe has no display.
Storage, including mass storage. Picaxe has none. The commodore uses storage media that are hard to get and incompatible with modern formats. Maybe a draw on that one (the 'standard' these days is probably an sd card). Is there a FAT32 driver for an sd card for the picaxe?
Serial ports - commodore and picaxe both have these. I'm not sure how the buffers work for both.
Printer port - possibly a draw.
Analog and digital input/output. Picaxe wins there hands down.
Current consumption - picaxe wins.
These are interesting questions. Retro computing is something I have been playing around with for a couple of years now
http://smarthome.viviti.com/propeller
We are looking at doing the Commodore 64 as we have an emulation now for the 6502. Just need the glue logic for the display/keyboard.