With those many websites that compare flights we usually have the opportunity to chose between different paths to reach the final destination.
Should the type of plane influence the decision? Ex. If I see that one company uses a Boeing 787 instead of a Boeing 767 or an Airbus A330-300, does that influence the travel experience in a perceivable way (In terms of turbulence attenuance, noise reduction, smoothness of flight, different types of landing/take off)?
As a general rule, not really, though there may be some factors that could cause you to consider it as part of your decision-making.
Some short-haul routes may be served by aircraft of significantly different sizes. For instance, some flights between San Francisco and Los Angeles may be on a Boeing 737, while others are on a CRJ Regional Jet. Still others may be offered on a turboprop like the Dash 8 or the even smaller Beechcraft 1900 (or even smaller aircraft, especially serving remote areas). Even those who aren't afraid of flying may be uncomfortable on small "puddle jumper" aircraft, as the seats and headroom are usually physically small and they can be more impacted by turbulence. Such aircraft can also require standard carry-on bags to be checked at the gate (they are usually returned to you at the gate on arrival), as lack large overhead bins.
The Boeing 787 Dreamliner can be pressurized to a lower cabin altitude and has a higher level of humidity than is typical. Efforts were made to lower cabin noise as well. This can lead to a more comfortable flight. Many of these enhancements may be carried over into future next-generation aircraft.
There may be differences in the cabins depending on the aircraft model, especially in premium classes. For instance, first class passengers on Lufthansa may have a separate bed on Boeing 747 aircraft, while those on Lufthansa's A340 will have to make do with sleeping on bedding atop their luxurious seat. Emirates business class is supposedly far better on the A380 (which includes a bar in the back) than the 777, though which you get mostly depends on your route. In other cases, an airline may be rolling out a new premium seat design, which is only available on some aircraft types. While the differences are less pronounced in economy, there may be variations in seat pitch (legroom) and configuration (2-5-2 vs 3-4-3 for example) that can make a significant difference in comfort.
Some people prefer newer aircraft. This can be difficult to guarantee by type alone though, as many aircraft have been produced over a period of decades. Even older aircraft have to go through a rigorous inspection and maintenance procedure though.
That said, most people are choosing their flights based on factors like price, route/time, and frequent flyer membership. Type of aircraft is usually pretty low on the priority list unless you're an enthusiast determined to fly a certain new model.
Also note that aircraft type is never guaranteed, and carriers reserve the right to make substitutions for operational reasons without notice.
