Intuitively, I though that $-1^2$ and $(-1)^2$ were exactly the same thing; however, it seems I was wrong as Wolfram Alpha (and any other calculator) returns $-1$ for the first case and $+1$ for the second one.
What is the difference between the two?
Your calculator (and Alpha) are correct: $-1^2 = -(1^2) = -1$ whereas $(-1)^2=+1$.
When evaluating expressions with multiple operations you have to follow the proper order of operations (order of precedence).
Quickly: Exponentiation then Multiplication/Division then Addition/Subtraction.
The reasoning behind this ordering has to do with the complexity of each operation (which ones are built from which other ones).
The most basic operation is succession (let's use "S" to denote successor). Then $S(n)=n+1$. Concretely $S(0)=1$, $S(1)=2$, etc. The inverse operation is declination (go down one): $D(n)=n-1$ so that $D(5)=4$ etc.
Addition is built from repeated succession: $3+2 = S(S(3))=5$ (start at $3$ and then go up twice). Subtraction is undoes addition. It can be realized as repeated declination: $3-2 = D(D(3))=1$ (start at 3 and go down twice).
Next, multiplication is built from repeated addition: $3 \cdot 2 = 3+3=6$ and division is built from repeated subtraction: $12/4 = 3$ since $12-4-4-4=0$ (three 4's to eliminate 12).
Finally, exponentiation is built from repeated multiplication: $2^3 = 2 \cdot 2 \cdot 2 = 8$.
Thus increment and decrement repeated give addition and subtraction. Addition and subtraction repeated yield multiplication and division. Multiplication repeated yields exponentiation.
This leaves us with our standard order of precedence.
So finally, $-1^2$ involves an exponentiation and a subtraction (negation is just a special case of subtraction -- subtract from zero). So exponentiation is "more important" (higher up on our hierarchy of operations) so it's done first: $1^2=1$. Then comes negation: $-(1^2)=-1$.
Of course, parenthesis override precedence (we must compute expressions within parentheses first), so $(-1)^2$ says to negate 1 first: $-1$. Then square: $(-1)^2 = (-1) \cdot (-1) = 1$.
One thing that we do in mathematics is to take a concepts, and flip between several interpretations of that concept. For example, is $\frac{45}{9}$ the number of equal partitions of of size $9$ that can made from $45$ things , or is it the size of each partition, given that there are $9$ partitions. Or perhaps it is the size of one thing divided by 9 and then replicated $45$ times.
Another example occurs with the expression $2-5$. Are we subtracting $5$ from $2$, or are we adding $-5$ to $2$. In any event, I don't know how to subtract $5$ from $2$, so instead, I'm going to reinterpret the expression as the negation of the negation of itself. $$-(-(2-5)) = -(5-2)$$ which, after applying the distributive property, I find much easier to evaluate.
So I have all of these interpretations that I like to flip back and forth between, but there on thing that must remain constant. Each interpretation must be consistent with every other interpretation. In the case of $-1^2$, I could have the following interpretations. $$\begin{array}{ll} \text{unary negation}&-1^2\\ \text{subtraction}&0-1^2\\ \text{multiplication}&(-1)\cdot1^2\\ \end{array}$$ While the first construct may seem ambiguous, we would like it to be consistent with the other interpretations, so exponentiation is given a higher priority than negation, for this reason.
