Huge difference.

As the name implies, a double has 2x the precision of float^[1]. In general a double has 15 decimal digits of precision, while float has 7.

Here's how the number of digits are calculated:

double has 52 mantissa bits + 1 hidden bit: log(2⁵³)÷log(10) = 15.95 digits

float has 23 mantissa bits + 1 hidden bit: log(2²⁴)÷log(10) = 7.22 digits

This precision loss could lead to greater truncation errors being accumulated when repeated calculations are done, e.g.

float a = 1.f / 81;
float b = 0;
for (int i = 0; i < 729; ++ i)
    b += a;
printf("%.7g\n", b); // prints 9.000023

while

double a = 1.0 / 81;
double b = 0;
for (int i = 0; i < 729; ++ i)
    b += a;
printf("%.15g\n", b); // prints 8.99999999999996

Also, the maximum value of float is about 3e38, but double is about 1.7e308, so using float can hit "infinity" (i.e. a special floating-point number) much more easily than double for something simple, e.g. computing the factorial of 60.

During testing, maybe a few test cases contain these huge numbers, which may cause your programs to fail if you use floats.

Of course, sometimes, even double isn't accurate enough, hence we sometimes have long double^[1] (the above example gives 9.000000000000000066 on Mac), but all floating point types suffer from round-off errors, so if precision is very important (e.g. money processing) you should use int or a fraction class.

Furthermore, don't use += to sum lots of floating point numbers, as the errors accumulate quickly. If you're using Python, use fsum. Otherwise, try to implement the Kahan summation algorithm.

^{[1]: The C and C++ standards do not specify the representation of float, double and long double. It is possible that all three are implemented as IEEE double-precision. Nevertheless, for most architectures (gcc, MSVC; x86, x64, ARM) float is indeed a IEEE single-precision floating point number (binary32), and double is a IEEE double-precision floating point number (binary64).}

The default choice for a floating-point type should be double. This is also the type that you get with floating-point literals without a suffix or (in C) standard functions that operate on floating point numbers (e.g. exp, sin, etc.).

float should only be used if you need to operate on a lot of floating-point numbers (think in the order of thousands or more) and analysis of the algorithm has shown that the reduced range and accuracy don't pose a problem.

long double can be used if you need more range or accuracy than double, and if it provides this on your target platform.

In summary, float and long double should be reserved for use by the specialists, with double for "every-day" use.

float (the C# alias for System.Single) and double (the C# alias for System.Double) are floating binary point types. float is 32-bit; double is 64-bit. In other words, they represent a number like this:

10001.10010110011

The binary number and the location of the binary point are both encoded within the value.

decimal (the C# alias for System.Decimal) is a floating decimal point type. In other words, they represent a number like this:

12345.65789

Again, the number and the location of the decimal point are both encoded within the value – that's what makes decimal still a floating point type instead of a fixed point type.

The important thing to note is that humans are used to representing non-integers in a decimal form, and expect exact results in decimal representations; not all decimal numbers are exactly representable in binary floating point – 0.1, for example – so if you use a binary floating point value you'll actually get an approximation to 0.1. You'll still get approximations when using a floating decimal point as well – the result of dividing 1 by 3 can't be exactly represented, for example.

As for what to use when:

• For values which are "naturally exact decimals" it's good to use decimal. This is usually suitable for any concepts invented by humans: financial values are the most obvious example, but there are others too. Consider the score given to divers or ice skaters, for example.

• For values which are more artefacts of nature which can't really be measured exactly anyway, float/double are more appropriate. For example, scientific data would usually be represented in this form. Here, the original values won't be "decimally accurate" to start with, so it's not important for the expected results to maintain the "decimal accuracy". Floating binary point types are much faster to work with than decimals.