Generally speaking, Saiyans have been shown to be the more formidable warriors when compared to Namekians. The latter is more focused on peace and in tune with nature. Saiyans, by contrast, are ruthless conquerors who grow strong with every defeat. For this reason alone, Vegeta was born with an advantage over Piccolo in terms of innate ability.
Saiyans are tough but their ability to heal has nothing on a Namekian. Throughout the series, Piccolo and other Namekians were able to regenerate limbs within a matter of moments. The only thing they seem to be unable to regenerate are their vital organs. This is seen early on in Dragon Ball Z when Radditz kills Piccolo with an energy wave.
The energy of Namekians does appear to be finite, like most organic, non-androids in the series. So Vegeta would not lose a battle of stamina. However, he won't be able to rely on severing or damaging limbs to achieve victory.
Spoilers ahead for the Dragon Ball Super Heroes movie!
The full-blooded Saiyans, Goku, and Vegeta have constantly broken limits and achieved numerous new forms. Piccolo has largely been left out when it comes to new levels of power and forms. This is of course if you don't count Orange Piccolo from Dragon Ball Super.
To the best of my knowledge, this form is not considered canonical so it wouldn't be a factor if Vegeta and Piccolo fought. As such, Piccolo would be fighting at the level he was at toward the end of the Tournament of Power arc against Ultra Ego Vegeta (highest level as of August 2022).
As far as on-screen battles go, Vegeta has the edge over Piccolo due to his main character status. However, given that Piccolo presumably has the knowledge of Kami since they fused he may know a bit more about battle. Kami/King Piccolo were around about 300 years prior to the Dragon Ball series beginning.
Piccolo also fused with Nail on Namek so it's likely that he gained more battle knowledge in doing so. Vegeta is an intelligent warrior in his own right, having been groomed since birth for combat but has not lived long enough to know as much as Piccolo.
I know, this has nothing to do with them fighting but being a father myself, I do love seeing father-son moments in Dragon Ball. Vegeta, although a bit cold-hearted, cares deeply for his family and has brief moments where he is somewhat affectionate toward them.
Piccolo is basically Gohan's father. If Dragon Ball took place in America, Goku would owe like 2 million Zenny in child support by now. Every since the Saiyan saga in Dragon Ball Z, Piccolo has provided advice, training, and mentorship to Gohan while Goku was out of the picture.
He has even played the role of Grandpa to Pan, often picking her up from daycare and training her (at least in Dragon Ball Super Heroes). Overall, I'd say Piccolo wins the better father award. It takes a special kind of guy to raise another guy's kid, especially one of a completely different species.
If we are strictly talking about a battle based on the current state of the manga or even the anime, Vegeta outclasses Piccolo in terms of speed and power severely. Their levels of power are not close enough for a decent battle. In fact, Vegeta may not even have to power up past Super Saiyan God Red.
Piccolo' Orange Transformation increases his power exponentially and enables him to best Gamma in combat after losing handily to him in their initial meeting. Piccolo is also able to land powerful blows on Cell Max who outclasses the original cell in terms of raw power. It has been said that even Broly would be unable to defeat Cell Max in his complete form.
So it stands to reason that Vegeta, who was unable to defeat Broly on his own would also lose to Cell Max. In the Dragon Ball Super Movie world, Vegeta and Piccolo are much more evenly matched than in the anime and manga. With no other point of comparison other than that they would both probably lose to incomplete Cell Max, I'd give the win to Piccolo.
Orange Piccolo wins due to his ability to regenerate, increase his size drastically and his higher battle IQ. Even though the size increase doesn't boost his power, it would force Vegeta to re-think his strategy in the middle of battle. Fighting a 7 foot tall enemy is a lot different than battling one that is well over 40 feet.
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