What is the difference between parseInt() and Number()?

How do parseInt() and Number() behave differently when converting strings to numbers?

Well, they are semantically different, the Number constructor called as a function performs type conversion and parseInt performs parsing, e.g.:

// parsing:
parseInt("20px");       // 20
parseInt("10100", 2);   // 20
parseInt("2e1");        // 2

// type conversion
Number("20px");       // NaN
Number("2e1");        // 20, exponential notation

Also parseInt will ignore trailing characters that don't correspond with any digit of the currently used base.

The Number constructor doesn't detect implicit octals, but can detect the explicit octal notation:

Number("010");         // 10
Number("0o10")         // 8, explicit octal

parseInt("010");       // 8, implicit octal
parseInt("010", 10);   // 10, decimal radix used

And it can handle numbers in hexadecimal notation, just like parseInt:

Number("0xF");   // 15
parseInt("0xF"); //15

In addition, a widely used construct to perform Numeric type conversion, is the Unary + Operator (p. 72), it is equivalent to using the Number constructor as a function:

+"2e1";   // 20
+"0xF";   // 15
+"010";   // 10

typeof parseInt("123") => number
typeof Number("123") => number
typeof new Number("123") => object (Number primitive wrapper object)

first two will give you better performance as it returns a primitive instead of an object.