As stated in my answer to Would we have more than 8 minutes of light, if the sun "went out"? , the Sun would cool at roughly constant luminosity for some tens of millions of years. That means "we" would not notice anything change on human timescales (apart from the lack of neutrinos, which has no measurable effect on anything apart from the odd atom in a vast vats of cleaning fluid or water, underground in South Dakota and Japan).
If $dL/dt$ was roughly $-L_{\odot}/30$ in solar luminosities per million years, then it would be 1000 years before the luminosity of the Sun fell by 0.1%, which is roughly the kind of variations we see over the course of the solar cycle.
However, the most likely evolution of the Sun is for it to contract, maintain a stable luminosity and compensate by increasing its surface (photospheric) temperature.
Given that luminosity is provided by gravitational contraction and the rate of change of gravitational potential energy, then
$$ L_{\odot} \simeq -\left(\frac{GM_{\odot}^2}{R^2}\right)\frac{dR}{dt}, \tag*{(1)}$$
where $R$ is the radius.
If the luminosity is constant and given by
$$L_{\odot} = 4\pi R^2 \sigma T^4,$$
where $\sigma$ is the Stefan-Boltzmann constant and $T$ is the effective (photospheric) temperature, then
$$ \frac{dL_{\odot}}{dt} = 4\pi \sigma \left(2R T^4 \frac{dR}{dt} + 4R^2 T^3 \frac{dT}{dt}\right) = 0 \tag*{(2)}$$
Combining equations (1) and (2) we find
$$\frac{dT}{dt} = \frac{T R L_{\odot}}{2 GM_{\odot}^2} = 90\ {\rm K/Myr}$$
i.e. The Sun surface would heat up by 90K every million years.
This gradual warming would change the spectral type of the Sun, moving it towards an F-type star. The spectrum of light it emitted would shift to shorter wavelengths (according to Wien's displacement law, the peak wavelength in the spectrum is inversely proportional to temperature) and a greater fraction of the Sun's light would appear in the ultra-violet part of the spectrum.
