Strictly speaking, a mantissa is part of a logarithm (specifically a common logarithm). For example, the common (base-10) logarithm of 1234 is 3.09132. From that number 3.09132 we can immediately see that we're dealing with a number between 103 = 1000 and 104 = 10000. And then the .09132 part essentially gives us the rest of the digits.
When working with such logarithms, .09132 is the mantissa and 3 is the characteristic. This is useful because multiplications and divisions by 10 change the characteristic but not the mantissa. For example, we immediately know that the logarithm of 123.4 is 2.09132, and that the logarithm of 1.234 is 0.09132.
In fact, back in the day when people used log tables to do higher-order arithmetic, those log tables didn't bother to give you the logarithm of every number. They just told you that 1234 had a mantissa of 09132, and it was your job to figure out the characteristic, based on whether you knew you started with 1234, or 1.234, or 1234000, or some other magnitude of number. (There was also a funny convention for numbers less than 1. Mathematically, the logarithm of 0.1234 is -0.90868, which doesn't have .09132 in it, until you realize that 0.09132 - 1 = -0.90868. Using the log tables, you said that the logarithm of 0.1234 had a mantissa of .09132 and a characteristic of -1, which was often written as 1̅.09132. If your browser didn't render that correctly, there's an overbar over the leading 1, indicating that it's negative, but the mantissa is still positive.) You can read more about this at the Wikipedia article on common logarithm.
So that's the story behind "mantissa". The significand, on the other hand, is simply one part of the representation of a number in scientific notation (aka exponential notation). If we say 1234 = 1.234 × 103, we've got a significand of 1.234 and an exponent of 3. In IEEE 754, in binary, if we have 57.375 =
0b1.11001011 × 25, we've got a significand of
0b0.11001011 without the implicit leading 1 bit) and an exponent of 5.
So a mantissa is something you can add an integer to in order to get the logarithm of the number you're interested in; it's the fractional part of the logarithmic representation of a number. A significand is something you can multiply by a power of the base in order to get the number you're interested in; it's the normalized part of a scientific or exponential notation.
So mantissa and significand really are two pretty different things. It's unfortunate, I suppose, that Burks started using "mantissa" in the context of floating point, because that term really caught on, and it's what pretty much everyone learns.
This is also explained in the "Terminology" section of the Wikipedia article on Significand.