As
newarts said, pictures taken with the same aperture and shutter (with everything else [ISO, light, angle, distance, etc] being equal) will look the same, no matter which modes got you there. And pictures taken with the same EV (combined effect of ISO, aperture and shutter) will be exposed the same but will have different DOF and motion blurring -- again, the modes that got you there don't matter to the captured image.
The shooting modes are just various paths to the same end. They're like descending in a tall building; you can take escalator, elevator, stairs, slide down a pole or rope, use rock-climbing techniques or a spiderman-web, or just jump out a window (with or without a parachute). They all take you to the same place.
P mode is a gateway combining Av and Tv modes. Try this: In P mode, aim at something. Take a meter reading by half-pressing and releasing the shutter button. Now spin the eDial to change the settings. Look through the viewfinder, or at the top LCD screen; note that shutter speed and aperture change in tandem to stay at the same EV. The light captured remains the same at all those settings, but the images will have different sharpness/blur, DOF, bokeh, all that stuff. You're invisibly changing modes.
EACH OF THOSE SETTINGS IS A DIFFERENT SOLUTION TO THE PROBLEM OF GRABBING THE IMAGE. Every shot is a problem to be solved; that's the fun here!
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EDIT/addendum: Earlier I mentioned using M mode in consistent light even with AF lenses. Here is an example.
I was in the Red Rock country around Sedona Arizona, shooting under clean blue skies with white puffy clouds. The 'scape was red rocks, tawny sand, plants of various shades of green, adobe and wood buildings, that sort of stuff. If I had been in an auto mode, every slight shift of angle would result in a different exposure. So I did some quick test metering and shooting and chimping until I found the setting that provided the best results, the truest colors. I left it at that setting for 2-3 hours of shooting 'scapes with the same zoom. Consistent light + consistent setting = consistent results.
And I could have done that without even metering, by using the
SUNNY 16 RULE. Remember my spiel about the ancient Kodak Retina 1 folder, back in post#16? That's a technique I used with it. No chimping allowed then! And yet it worked pretty damn well.
See also
WHY YOUR LIGHT METER LIES TO YOU.