Depends if you think you will ever get more serious into photography and get a new camera one day
If I were to guess, most likely you want to learn more (or you wouldn't be even asking here ) so you can improve your photography
The short answer for your Coolpix P series not going to make a big difference . The reason why is your camera has VERY limited aperture available ( and limited usable iso range )
For example its about 3 to 8 at the very wide end . Telephoto is only like 6 to 8 !
Therefore when you are in Aperature priority, as you change from 6 to 8 on the telephoto end , not a really big difference
A range of 3 to 8 is much more noticeable, but unfortunately you camera can only do this the wide end so , again less noticeable since depth of field a function of BOTH aperture and focal length
Aperture Priority
As already mentioned most serious enthusiast who are not shooting in Manual will be shooting in Aperture priority
The reason ( already mentioned ) is that the aperture makes a huge difference on the composition ( due to the depth of field ) that you are trying to convey. Therefore you want the control over it
For example, when shooting portraits @ 85 or 100mm or longer, there is a huge difference in the "look" between F2.8 , F4 and F11 . If I want everything to be sharp I shoot @ F11 as a go to. Depending on distance to subject, @ F 1.4 , to 2.8 you can have eyes sharp but ears not
The purpose of this is to draw the attention to what is in focus . A portrait with a shallow depth of field IMO looks much better than verything sharp . Just look at any wedding photographers portfolios
When shooting a portrait , the shutter speed even as low as 1/60 all the way to insanely fast will have no bearing on the composition
If you google image examples of depth of field you will appreciate why having control of Aperature is very very important
Shutter Priority
Unless I am goofing around trying to "drag the shutter" or trying to stop motion when using ambient light don't really pay any mind to the shutter speed . When using flash / strobes, the flash "freezes" the motion since the ambient contributes very little . Exception is when using flash for fill light
Better cameras have the ability to have "shutter speed range limits" . For example, even though I generally shoot Aperture priority , i have my camera set for certain minimum shutter speeds so as the camera does its own adjustment it hits a wall
This ensures I do not have to slow a SS for certain situations ( the camera then starts ramping ISO after it hits wall off minimum SS )
Manual Mode
If I am not shooting in Aperture priority I am shooting in manual .
This is generally when I'm using studio flashes / strobes
Where you have to be careful is many cameras still do a Auto Iso even though you are in Manual ! Go figure