Fighting Skill: Jon Snow vs. Ramsay Snow (Spoilers Extended)

We all saw the Battle of the Bastards on the show, and while visually pleasing if it happens it will likely happen much different in the books.

If interested: The Stark Direwolves vs. Ramsay's Hounds

If we compare Ramsay's skill level:

"He fights for you," Reek blurted out. "He's strong."
"Bulls are strong. Bears. I have seen my bastard fight. He is not entirely to blame. Reek was his tutor, the first Reek, and Reek was never trained at arms. Ramsay is ferocious, I will grant you, but he swings that sword like a butcher hacking meat." -ADWD, Reek III

with Jon's (not even taking into account valyrian steel):

"Older and bigger and stronger, that's the truth. I**'ll wager your master-at-arms taught you how to fight bigger men at Winterfell, though. Who was he, some old knight?"**
"Ser Rodrik Cassel," Jon said warily. There was a trap here. He felt it closing around him.
Donal Noye leaned forward, into Jon's face. "Now think on this, boy. None of these others have ever had a master-at-arms until Ser Alliser. Their fathers were farmers and wagonmen and poachers, smiths and miners and oars on a trading galley. What they know of fighting they learned between decks, in the alleys of Oldtown and Lannisport, in wayside brothels and taverns on the kingsroad. They may have clacked a few sticks together before they came here, but I promise you, not one in twenty was ever rich enough to own a real sword." His look was grim. "So how do you like the taste of your victories now, Lord Snow?" -AGOT, Jon III

but also keep in mind Jon is super strong as well:

"My lord is wise." Ser Alliser seized Jon by the arm.
Jon yanked away and grabbed the knight by the throat with such ferocity that he lifted him off the floor. He would have throttled him if the Eastwatch men had not pulled him off. Thorne staggered back, rubbing the marks Jon's fingers had left on his neck. "You see for yourselves, brothers. The boy is a wildling." -ASOS, Jon IX

even if he is considered the quicker of the Snow/Stark boys:

"No," Jon Snow said quietly. "It was not courage. This one was dead of fear. You could see it in his eyes, Stark." Jon's eyes were a grey so dark they seemed almost black, but there was little they did not see. He was of an age with Robb, but they did not look alike. Jon was slender where Robb was muscular, dark where Robb was fair, graceful and quick where his half brother was strong and fast -AGOT, Bran I

We also see Jon constantly training/improving/fighting throughout the series:

• Alliser Thorne/trainees
• Wights
• Wildlings
• Qhorin Halfhand (with Ghost)
• Iron Emmett
• Rattleshirt (Mance)

and while Ramsay has experienced battle/conflict (burning Winterfell/retaking Moat Cailin) we get no real mention of true hand to hand combat (he does take off Rodrik's arm).

It should also be noted that if this does take place, Jon will likely be undead (and therefore possibly super focused on Ramsay seeing as that was his focus before he died) and a valyrian sword.

The logistics of this happening are also interesting as we know that Ramsay is headed out to meet Stannis outside Winterfell. We know Stannis survives in order to burn Shireen, so Ramsay either is defeated and flees (back to WF or the Dreadfort), defeats Stannis and pursues him to the Wall, etc. etc. But the point is that it is snowing hard (hard to travel) and Jon Snow is dead* (for the watch).

TLDR: If Jon/Ramsay end up fighting each other, Jon is much more skilled as a fighter.