At a high level, intrinsics is something like a different syntax for vanilla SIMD asm. For example to add two 128 bit width vectors you can use _mm_add_ps() (thats leads to a SSE instructions).
The advantage of using intrinsic is that both, gcc and icc understand them. No need to #ifdef __GCC__ and other compile-time hacks. On the other side, i386 intrinsics are limited. For every SIMD extension (mmx, sse, ...) you must implement a own specialized version via
#ifdef __SSE2__ [...] #else # error Programmed Error #endif
If you use it with gcc, gcc will instantly expand them to __builtin_ia32_addps - a gcc built-in (search the gcc header search path for xmmintrin.h - type gcc -print-search-dirs if you doesn't know the current search paths).
The other opinion is to let the compiler optimize your code. But often the code is to complex that the compiler isn't able to do this task!
At the end: if you MUST use SIMD extension you SHOULD use intrinsics. They are more portable, more readable compared to asm and the introduced loss of control, also compared to vanilla asm is evanescent humble.