Who decides how to "handle" an aircraft, with regards to severity of manoeuvres etc?
Pilot in command, a.k.a. captain, has the last say in how the aircraft will be handled. However most of their decisions are driven by something:


Obviously performance of the aircraft. This is affected by weight (both cargo and fuel; aircraft only take the amount of fuel they need for the route they are flying and it makes a big difference) and weather (mainly temperature and head-wind).
Standard operating procedures (SOP). Certain operations can indeed be done in more than one way and each airline may indeed make slightly different choices. These are reflected in the recurrent pilot training.
Route flown. Airline flights are carried out under instrument flight rules, so the route is assigned by the air traffic control.



  Take off seemed more "brutal", and from a position slightly more at the beginning of the runways.


There is some freedom in choosing the take-off setting and standard operating procedure of different airlines may make different choices. Usually if the runway is long enough the engines are not set to full thrust, but so called “flex” thrust that is calculated sufficient for the take-off to limit wear of the engines.

What thrust and flap setting are possible is determined by weight, runway length, wind and temperature, but within the possible range, different airline may have different preference in their SOP (higher power or less flaps for faster climb v.s. lower power for less engine wear and more flaps for lower gear wear).

Noticeable difference might be in using “rolling take-off” where power is added immediately after turning on the runway without stopping there first. Whether it is possible depends on ATC, but some airlines seem to never do it.


  After take off, the airplanes did a lot of maneuvers during climbing. While EasyJet's planes usually do a variation of "turn, long climb, another turn", the Vuelings' did "a lot" of turns during climbing.


This depends on the departure route assigned by the controller. Note that even if flying the same route, the departure route could be different if different runway was used or if different routing was selected due to congestion (European airspace gets very busy at times and then the plane may wait or use different route).


  Landings, from the point of view of turning, are the same (as it should be).


Not necessarily. Arrival routes may differ similarly to how departure routes do. In fact even more so, because on arrival the aircraft are often vectored, that is get ad-hoc headings assigned by controller instead of flying standard route depicted on the chart, to achieve appropriate separation. On take-off the aircraft simply wait on the ground until the previous aircraft is far enough so it's not needed often.


  But after touchdown...it seemed to me the pilot kept an higher speed for much much more time, so that in both cases after the gear touching ground the plane reached the apron in a matter of seconds


Again there can be differences in SOP regarding preferred auto-brake setting, but usually every pilot will want to vacate the runway as soon as possible, so it seems more like difference in what exits were available.

It might have also been difference in the aircraft. There was some talk about Airbus implementing “brake to vacate”, where the computer handles the brakes so the aircraft slows down just before the next suitable turn-off, on the new A320, but I found no reference whether they already did.


  and in both case the planes were swinging a lot.


The vertical tail acts like a weather-vane and the thrust reversers reduce lateral stability, so the aircraft always needs some directional corrections during landing.

In the air the aircraft turns by banking, so you don't feel any lateral movement. But on the ground it can't and due to the speed you feel even very slight turns.