Yes, the periods of high solar activity (i.e. more sunspots) are also the periods of increased brightness.
The main reason for this is that although the spots themselves are indeed darker and thus radiate less energy, they are always accompanied by wide areas of increased brightness around them - so-called faculae, which results in the net increase of irradiance.
Although the solar constant (i.e. the total electromagnetic energy flux from the Sun per unit area) changes not by much per cycle of activity (changes of the order of 0.1%), the flux in certain wavelengths might be significantly altered. For example, in 1-65 nanometer range (UV) the flux at solar maximum (many spots) might be about 7 times bigger than at minimum. X-ray flux is also significantly lower at solar minimum, and the same is true for radiowaves (the flux at the wavelength of 10.7 cm is actually one of the main indices of solar activity). The shorter wavelengths play a major role in the dynamics and chemistry of our upper atmosphere. It is well known in astronautics, for example, that the lifetime of a satellite in a low-Earth orbit during solar maximum could be significantly less than at solar minimum because the extra energy from the Sun heats the atmosphere up and expands it, increasing the drag that leads to faster orbital decay. One famous example was the Skylab space station, for the lifetime of which the predictions without the correct account of solar activity gave the error of about 4 years.
