Yes.
Different brand tyres will feel different. But a lot will depend on the model, as one brand of tyre may work well on a Panda, but be rubbish on a Punto.
Post the question in the relevant forum for your model and let others give their recommendations and experiences.
As an example, my Panda runs 155/80 13. The Continental tyres that were on it when I bought it were a little noisier and less grippy than expected. Surprising for a good brand, but might be due to age too. The fronts have been replaced with Vredestein and are quieter, softer ride and grip better.
However, raising the pressures has also made a significant difference.
Go to the website for your tyre manufacturer. Try to find, often well hidden, their listing for pressure recommendations. Most of the good brands will have this.
Both Continental and Vredestein show higher pressures than Fiat give in the handbook. The higher pressures make a significant difference, with better grip, quieter running, and surprisingly, a more comfortable ride.