Why can't we define a variable inside an if statement?

Try C#7's Pattern Matching.

Using your example:

if (new StringBuilder("test") is var sb && sb != null) {
    Console.WriteLine(sb);
}

This is because section 8.5.1 of the C# language spec. states:

Furthermore, a variable initializer in a local variable declaration corresponds exactly to an assignment statement that is inserted immediately after the declaration.

This basically means that, when you do:

StringBuilder sb = new StringBuilder("test")

You're, in effect, doing the exact same thing as:

StringBuilder sb; sb = new StringBuilder("test")

As such, there is no longer a return value for your check against != null, as the assignment isn't a single expression, but rather a statement, which is a local-variable-declarator comprised of an identifier followed by an expression.

The language specification gives this example, stating that this:

void F() {
   int x = 1, y, z = x * 2;
}

Is exactly equivalent to:

void F() {
   int x; x = 1;
   int y;
   int z; z = x * 2;
}

This has to do with the difference between a statement, and an expression. An expression has a value, whereas a statement does not.

Using your examples, notice these classifications:

StringBuilder sb; // statement

sb = new StringBuilder("test") // expression

StringBuilder sb = new StringBuilder("test"); // statement

Notice that only the middle portion is a expression.

Now we move onto your conditional statement. The syntax for using the not-equals operator is

expression != expression

So on both sides of the != you need something that actually has a value (this just makes sense). Ergo, you cannot have statements on either side of the operator. This is why the one version of your code works, while the other does not.