What are the advantages and disadvantages of using one instead of the other in C++?
If you want to know the true answer, you should read What Every Computer Scientist Should Know About Floating-Point Arithmetic.
In short, although
double allows for higher precision in its representation, for certain calculations it would produce larger errors. The "right" choice is: use as much precision as you need but not more and choose the right algorithm.
Many compilers do extended floating point math in "non-strict" mode anyway (i.e. use a wider floating point type available in hardware, e.g. 80-bits and 128-bits floating), this should be taken into account as well. In practice, you can hardly see any difference in speed -- they are natives to hardware anyway.
Unless you have some specific reason to do otherwise, use double.
Perhaps surprisingly, it is double and not float that is the "normal" floating-point type in C (and C++). The standard math functions such as sin and log take doubles as arguments, and return doubles. A normal floating-point literal, as when you write 3.14 in your program, has the type double. Not float.
On typical modern computers, doubles can be just as fast as floats, or even faster, so performance is usually not a factor to consider, even for large calculations. (And those would have to be large calculations, or performance shouldn't even enter your mind. My new i7 desktop computer can do six billion multiplications of doubles in one second.)