Response of a system to a step function (heaviside)
Fat32's derivation of the result via the Fourier transform is correct, but I think that your original question hasn't really been answered ("what am I doing wrong?"). The real reason why WolframAlpha gives you a wrong result is the different definitions of the Fourier transform used by WolframAlpha and the one used by you to derive $H(\omega)$. By default, WolframAlpha uses this definition of the Fourier transform:

$$F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{j\omega t}dt\tag{1}$$

Apart from the factor $1/\sqrt{2\pi}$ (which makes the transform unitary), the actual problem lies in the sign of the exponent. You implicitly assume a negative sign, whereas WolramAlpha uses a positive sign.

There are two ways to solve the problem. Either you change Wolfram Alpha's default parameters using FourierParameters -&gt; {1, -1} as explained here, or you change $H(\omega)$ accordingly, which means that you would need to use

$$H(\omega)=\frac{1}{1-j\omega RC}\tag{2}$$

As an example I tried the second method, and WolframAlpha indeed provides the correct result.

One more thing, for this kind of problems it's usually more convenient to use the Laplace transform because this avoids the use of generalized functions, such as the Dirac delta.