Destroying the remains doesn't necessarily destroy the spirit
The show confirms this a few times in later seasons:
Ash dies in "All Hell Breaks Loose, Part 1" and, because of the manner of his death, it's likely that his remains were destroyed at the same time (or that he received a Hunter's Funeral shortly after). However, we see him again in the season 5 episode "Dark Side of the Moon."
The episode involves Sam and Dean infiltrating Heaven; Ash grabs them and takes them back to his personal afterlife
Two Hunter characters are killed in "Abandon All Hope...", in season 5.
It's Ellen and Jo Harvelle.
The characters are killed by an exploding gas can, which they are sitting next to; assuming there were any remains left after that, the Brothers presumably gave them a Hunter's funeral.
Both of these characters are later seen and/or referenced:
- One appears as a vengeful spirit in the season 7 episode "Defending Your Life", having been summoned by the god Osiris to punish Dean
- The other is referenced in the season 7 episode "The Mentalists", where a medium relays a message from them (in the afterlife) to Dean
Crowley has a line in the season 6 episode "Weekend at Bobby's" that also implies this:
Crowley: I loathe [Gavin, Crowley's son from when he was human]. You want to torture him, just let me pull up a chair and watch. Hell, burn his bones and send him down to me and we can have a family reunion.
Supernatural Season 6 Episode 4: "Weekend at Bobby's"
Late in season 7, a character dies, and is given a Hunter's funeral:
This character returns as a spirit starting just a few episodes later, tethered to one of his former possessions. The Brother banish him (by destroying the object in question) in the season 7 finale.
The character later reappears in the season 8 episode "Taxi Driver"; as part of the season-spanning macguffin plot, Sam has to go into Hell to retrieve his soul.
It seems as though a spirit whose bones are burnt just gets another visit from a reaper, and get ferried onto whatever afterlife they earned; this is the implication of a conversation with the rogue reaper Ajay in "Taxi Driver":
Ajay: I am the reaper who took [character from bullet point 4, above] to Hell.
Sam: [Character] in Hell? We burned his bones. Once we did that, it was over. End of story.
Ajay: Hmmm, not necessarily.
Dean: No, no, no, 'cause, see, [Character] was on the good side of things, and good guys go to the penthouse.
Ajay: Usually, mostly. Depends on who you know, what palms get greased. If you're on the King of Hell's no-fly list, no way you cruise the friendly skies.
Supernatural Season 8 Episode 19: "Taxi Driver"
It should be noted that there are other possible routes:
- Monsters, some of whom (consider vampires, werewolves) are just mutated humans go to Purgatory
- It's possible that some get sent to the Empty; Billie threatens Sam and Dean with this in her first appearance, but she never follows through so we don't know if she was serious or just trying to intimidate them. However, we have it on reasonably good authority (as of season 13) that demons go to the Empty
But neither of these really apply to John, so it's pretty clear what happened in his case.