The sides are linked together by seven bridges across the river, but the most famous is the Chain Bridge. This bridge hosts the stone lions that survived the devastation resulted by the World War II and the Battle of Budapest in 1945.
The Buda side of the city is geographically located in the Buda hills, being the classier and residential part of Budapest. For that reason, Buda is also quieter and the perfect place to go to enjoy an amazing sightseeing experience over the city. Buda is full of royal attractions, since the beautiful Fisherman’s Bastion and the Buda Castle. From this sites, the sightseeing has the best features across the river. Those combined with the thermal baths on top the hills and the presence of the Hungarian Liberty statue makes Buda a ‘must-to-visit’ for amazing experiences in Budapest.
Now, I think the two sides of the city are completely different and not comparable. If you like something calmer where you can enjoy the sightseeing among this amazing city – Buda is perfect. But if you enjoy something more agitated and social, then definitely Pest is the perfect side to go. But for a better understanding and to enjoy the full experiences that Budapest can provide, the both sides complement each other and deserve to be visited.