To echo the most salient points: If you are using this on a single compiler/HW platform as a software only construct, then endianness will not be an issue. If you are using code or data across multiple platforms OR need to match hardware bit layouts, then it IS an issue. And a lot of professional software is cross-platform, hence it has to care.
Here's the simplest example: I have code that stores numbers in binary format to disk. If I do not write and read this data to disk myself explicitly byte by byte, then it will not be the same value if read from an opposite endian system.
Concrete example:
int16_t s = 4096; // a signed 16-bit number...
Let's say my program ships with some data on the disk that I want to read in. Say I want to load it as 4096 in this case...
fread((void*)&s, 2, fp); // reading it from disk as binary...
Here I read it as a 16-bit value, not as explicit bytes.
That means if my system matches the endianness stored on disk, I get 4096, and if it doesn't, I get 16 !!!!!
So the most common use of endianness is to bulk load binary numbers, and then do a bswap if you don't match. In the past, we'd store data on disk as big endian because Intel was the odd man out and provided high speed instructions to swap the bytes. Nowadays, Intel is so common that often make Little Endian the default and swap when on a big endian system.
A slower, but endian neutral approach is to do ALL I/O by bytes, i.e.:
uint_8 ubyte;
int_8 sbyte;
int16_t s; // read s in endian neutral way
// Let's choose little endian as our chosen byte order:
fread((void*)&ubyte, 1, fp); // Only read 1 byte at a time
fread((void*)&sbyte, 1, fp); // Only read 1 byte at a time
// Reconstruct s
s = ubyte | (sByte << 8);
Note that this is identical to the code you'd write to do an endian swap, but you no longer need to check the endianness. And you can use macros to make this less painful.
I used the example of stored data used by a program.
The other main application mentioned is to write hardware registers, where those registers have an absolute ordering. One VERY COMMON place this comes up is with graphics. Get the endianness wrong and your red and blue color channels get reversed! Again, the issue is one of portability - you could simply adapt to a given hardware platform and graphics card, but if you want your same code to work on different machines, you must test.
Here's a classic test:
typedef union { uint_16 s; uint_8 b[2]; } EndianTest_t;
EndianTest_t test = 4096;
if (test.b[0] == 12) printf("Big Endian Detected!\n");
Note that bitfield issues exist as well but are orthogonal to endianness issues.