I know that if I buy an Intel Galileo EDB, I can download some separate software for it - can I use the standard Arduino IDE instead? And is there any major differences that I should know about the programming language? :~
Galileo can be programmed with the Arduino software (download). When you are ready to upload the sketch to the board, program Galileo through the USB Client port by selecting "Intel Galileo" as your board in the Arduino IDE
Why then is there the specific software? And are there any differences (such as new functions) in the programming language?
Galileo IDE will include an intel x86 compiler, and be modified to run it with the appropriate commands.
Similar to the way that Energia adds an MSP430 compiler, and MPIDE adds a PIC32 compiler.
You could combine them all into one (probably), but it would make for a really large download. (each compiler seems to be about 300-500MB.)
I'd expect the galileo compiler to support additional function; probably a full clib, and maybe additional c++ and x86-specific functions. Galileo has a lot more resources and memory than the AVR, so there was presumably less motivation to write "small and efficient libraries." Galileo probably has 32-bit "ints", for example.
(That said, I don't have a Galileo, so it's hard to say for sure. The whole PURPOSE of being "Arduino compatible" is to have the IDE and core functions look "about" the same (as much as possible.))