Manual 2 for the PICAXE, the command dictionary, mostly covers what is called a library of pre-written routines in the C world. Instead of commands, a c program would call a function in a library. Most microcontroller manufacturers provide libraries for their devices. The Arduino is particularly popular because its usual development environment has libraries of functions to work with most common peripherials just like the PICAXE commands. Many times you'll find people that sell 'shields' or Arduino compatible peripheral sensors and devices also provide a C library to make it easy to use it with the Arduino.
The differences in syntax are really rather minor. C tends to add punctuation to define scope or terminate a statement.
C provides a lot more nuance when it comes to the types of variables you can use. PICAXE is like assembly in that the variables you can use are based simply on the hardware it runs on. That means words and bytes and arithmetic is unsigned integer. In C, you have bytes, words, integers, floating point numbers, and signed arithmetic (all provided by built in libraries that are a part of the language definition). Note that mcu C compilers often fudge a bit on the standards for built in functions as the processors and memory just aren't there to do them right.
Microchip has a free C compiler and development environment for the PIC as used by PICAXE called MPLAB-X and their manuals for it are reasonably decent (see
http://www.microchipc.com/sourcecode/ ).
http://www.mikroe.com/products/view/11/book-pic-microcontrollers/ is a free book that covers the territory for beginners.
If you really want to go to town, check out
http://chipkit.net/ - about the same price as the Arduino but based on a 32 bit Microchip MCU with a free development environment and a lot of attention to Arduino compatability.
One of the other major issues to keep an eye on in using mcu's is loading the 'firmware' - PICAXE makes this easy as it includes what is called a bootloader to talk to a host PC and download your program. Other hobbyist mcu systems also usually have something similar. Without this, you need a special programming device to get code from your computer to the mcu.