The competition between Barcelona and Real Madrid will be even more competitive next season because both teams have made several efforts to bring in a number of players so that the fight for the La Liga title will be even tighter. Real Madrid will try to make several adjustments with their new coach while Barcelona will be much more solid because they are already well connected with the coach. La Liga next season will still be about rivalry and I am sure that both teams will compete for the title because if you look at it, there is no other team that might be able to match them in terms of fighting for the title.
Of the two teams you mentioned, I still think that Real Madrid will still have a little difficulty when they meet Barcelona again. Because the Real Madrid team still has to make adjustments with the tactics of their new coach for next season, while Barcelona already has quite extraordinary abilities plus new players who can help strengthen all lines in the team. So of the two teams, Barcelona might still be superior to Real Madrid because Alonso still has to make improvements to several players who are currently at Real Madrid.
It will be interesting to see whether Real Madrid, with all their changes, will be able to be better next season, or whether they will remain below Barcelona. Real Madrid really had a bad season last time out, where they not only failed to win a trophy but they always lost when faced Barcelona. Barcelona is not as rich as Real Madrid, and they also have limitations in buying new players because they don't have much budget left, but they have a solid squad, even though many of their players are still very young.
The transfer window is still open, and I think Real Madrid is likely to sign some more new players. While Barcelona seems to be focusing only on Nico Williams.