So this is something I've always been wondering.
So what is the difference between the Hobgoblin and the Green Goblin?
I honestly have not read a lot of the comics. Most of what I know is from the old Fox series. And I don't remember a lot of it.
From the comics:
They are two different villains. Hobgoblin was created because the writers didn't want to bring Green Goblin back to life in any way; so they created a new character that resembled him, but was different.
This is Hobgoblin. Τhere were many Hobgoblins, but the first Hobgoblin was Roderick Kingsley.
The Hobgoblin was created by writer Roger Stern and artist John Romita, Jr. for The Amazing Spider-Man #238 (March 1983). Like other writers Stern found himself under pressure to have Spider-Man fight the Green Goblin again, but did not wish to bring Norman Osborn or Bart Hamilton back from the dead, have Harry Osborn become the Green Goblin again, or create yet another Green Goblin. Instead he created a new character as heir to the Goblin's legacy and developed the Hobgoblin.
DeFalco, Tom (2004). Comics Creators on Spider-Man. Titan Books.
Whereas Green Goblin is the classic enemy of Spider-Man.
The Hobgoblin used the same formula that the Green Goblin used.
Using a modified version of the original Goblin formula used by Norman Osborn, Kingsley acquired superhuman strength, speed, stamina, and reflexes
Adding some detail to the accepted answer by Shevliaskovic.
Similarities between Green Goblin and Hobgoblin:
Differences: The most notable difference may be that most versions of the Hobgoblin have been sane, while most versions of the Green Goblin have been insane.
To provide a bit more detail on the characters from the comics:
The Green Goblin first appeared in AMAZING SPIDER-MAN #13 (July 1964). His origin and identity weren't revealed for a couple of years (issue #39-40).
The original Green Goblin was Norman Osborn. His mind was affected following a chemical accident at his company, basically causing him to have a split personality: Norman, the father who may have been a bit too occupied with his business, and the Goblin, the evil criminal mastermind. Before the revelation of his dual identity, Peter Parker and Norman's son Harry Osborn met at college, and eventually became best friends.
Over the next few years, Norman Osborn was mostly the busy businessman and father, with occasional returns to being the Goblin. In (what was at the time) his next-to-last appearance, seeing Harry in danger following a drug overdose shocked him from the Goblin back to Norman (issue #98, July 1971).
About 2 years later, Norman again became the Green Goblin (issues 121-122, June and July 1973). This was the story where he killed Peter Parker's then-girlfriend, Gwen Stacy (in 121), then inadvertently killed himself (he tried to attack Spider-Man by using his remote to steer his Goblin Glider, which was damaged and had a sharp piece of metal sticking out; Peter's spider-sense warned him something was up, and he dodged - which left Norman right in the path of the glider, which hit and killed him (or so we believed at the time)).
Following his father's death, Harry had a breakdown. His grief over his father, combined with his inadvertent discovery that Peter was Spider-Man (and, probably, Peter's incipient relationship with Harry's ex-girlfriend, Mary Jane Watson), combined to drive Harry crazy. He located a cache of his father's Goblin equipment, and attacked Peter as a new Green Goblin (issues #136-137, September and October of 1974). Harry was locked away for a short while, received therapy, and returned with no memory of Peter's identity.
Unfortunately, his therapist, Bart Hamilton, upon learning about all that happened with Harry, took advantage. He got Harry to reveal the location of one of his father's stashes of equipment, and became the third Green Goblin (issues 176-180, January-May of 1978). Under the circumstances, it's assumed that this is Harry until the end of the story. To the best of my knowledge, Hamilton has not been seen again (and certainly has not taken over the Green Goblin role again). Harry resumed his identity as the second Goblin long enough to stop Hamilton, then forgot about it again. (Note: this was when he fell in love with Liz Allan, his wife until his apparent death years later).
A quick sidebar: when Norman Osborn related the Goblin's origin, he notes that he was trying out a formula he'd taken from a former business partner, which exploded. He was apparently hospitalized for weeks, undergoing multiple surgeries to save his life. According to the surgeons, the explosion caused brain damage that was too deep for an operation to reach. Norman claimed he wasn't brain damaged: "... the accident made me more brilliant than I had ever been!" He also claimed that he was "... stronger ... smarter ... tougher than anyone else!"
In context, this is almost certainly his insanity talking. In his encounters with Spider-Man, he generally doesn't seem to be a physical equal to Spider-Man - he relies on weaponry to even the odds against Spidey's powers. While there's no direct reference in the Goblin's origin of him recieving super powers by the accident, in the Goblin's next appearance in Spectacular Spider-Man magazine issue #2 (1968) it is stated early in the story, while being the Green Goblin, Norman Osborn possesses maniacal strength. At some point during the story Osborn is seen throwing around some hefty looking furniture while searching for Peter Parker, with Parker noting "His strength is fantastic! He's juggling that furniture as if it's made of cardboard." That said, he is a maniac, so "maniacal strength" doesn't necessarily equate to super-human strength.
However, as we meet the Hobgoblin, these facts begin to change.
As noted in the accepted answer by Shevliaskovic, a few years later a new villain was introduced: the Hobgoblin. His costume was similar to the Green Goblin's, and he used the same weapons as Norman had created (Goblin Glider, pumpkin bombs, etc.). His identity was kept a secret for years (far more years than originally intended - the actual truth didn't come out until the following decade!). The Hobgoblin first appeared in AMAZING SPIDER-MAN 238 (March 1983), and was the focus of the title for about a year.
This was the point when the story first changed as to what happened in that explosion that sent Norman down the path to becoming a super-villain. It now appears that the formula did give Norman some sort of super-powers, plus caused brain damage - not due to the explosion, but due to some flaw in the formula itself.
The mystery man behind the Hobgoblin mask perfected the Goblin formula, so that he could take it without it exploding and without insanity as a default side-effect (though overexposure could still cause madness). He took it, giving himself the strength, speed, and agility to make him a direct physical threat to Spider-Man, even without the weaponry.
The Hobgoblin protected his identity on more than one occasion by putting someone else behind the mask. In a peculiar twist, the Hobgoblin was revealed to be Ned Leeds, husband of Peter's first crush, Betty Brant (AMAZING SPIDER-MAN 289, June 1987). Oddly enough, it was revealed after Ned was killed in another comic (SPIDER-MAN VS WOLVERINE, February 1987). A mercenary super-villain by the name of Jason Macendale, who had worked under the identity of Jack O'Lantern, took over the Hobgoblin identity.
Macendale had has own problems as the Hobgoblin. During the X-Men Inferno event, he's merged with a demon (SPECTACULAR SPIDER-MAN 147, February 1989), granting him more power (but, you know, merging him with a demon). Before too long, the whole "merged with a demon" part of the deal becomes problematic; he manages to separate himself from the demon - which goes on to retain a similar look and M.O. for several years as the Demogoblin:
Macendale resumed the Hobgoblin identity, and clashed with the Demogoblin multiple times. He was finally able to defeat the Demogoblin after getting access to Kraven the Hunter's strength formula (in SPIDER-MAN 47-49, June-August 1994). (Yeesh - makes you think that everyone in the Marvel U should have super-strength by now; there's enough of these formulas around). Shortly thereafter, Macendale gained cybernetic enhancements (SPIDER-MAN 68, May 1996).
However, these ultimately did him little good. You see, the original Hobgoblin wasn't Ned Leeds; it was actually a businessman named Roderick Kingsley. This was revealed in SPIDER-MAN: HOBGOBLIN LIVES 1-3 (January-April 1997). And when he was ready to come back, Kingsley wasn't thrilled about the usurper to his title - so, he killed Macendale in the first issue of that mini-series.
Meanwhile, the Green Goblin identity had mostly been unused. Harry used the Green Goblin outfit and weapons in a heroic capacity on a few occasions (such as WEB OF SPIDER-MAN 66-67, July and August of 1990); however, he eventually slipped back into insanity (aggravated by using a version of his father's formula), and (apparently) died in SPECTACULAR SPIDER-MAN 200 (May 1993).
Now, Norman has a lot of these caches of his weapons. A few years later, Phil Urich (nephew of Ben Urich, a Daily Bugle reporter originally seen in DAREDEVIL) stumbles across yet another one of these, and takes up the weapons to become a heroic version of the Green Goblin (GREEN GOBLIN 1-13, October 1995 - October 1996). He's able to gain the effects of the Goblin formula temporarily via a gadget, which avoids the psychotic effects of the actual formula. His career is cut short when his equipment is destroyed.
Then, not longer after that, we get the second big change regarding the original Goblin formula. Evidently, it comes with a healing factor. Norman Osborn survived being stabbed in the heart by his own broken glider. After spending years abroad, he came back and was revealed as having manipulated many of the events affected the lives of Peter, his *then) wife Mary Jane, and his clone, Ben Reilly (SPIDER-MAN 75, December 1996).
Following his return, Norman whitewashes his own past, convincing many that he never was the Green Goblin. He does resume the identity in his continuing efforts against Peter Parker.
Norman appears to be reasonably sane at this point. However, in an effort to gain more power, he helps pull together a ritual called the Gathering of Five. Unfortunately for him, power is one of five possible outcome for the five participants; he is driven mad (again) through the ritual (SPIDER-MAN 98, December 1998). He yet again regains enough of his sanity to be a threat shortly thereafter (starting in SPIDER-MAN: REVENGE OF THE GREEN GOBLIN 1-3 (October - December 2000), and following into AMAZING SPIDER-MAN 25 and PETER PARKER: SPIDER-MAN 25 (both January 2001)).
Norman continued to appear sporadically for a few years, winding up in jail multiple times, until Warren Ellis made him the head of the Thunderbolts program (at that time, a government "super-hero" team consisting of convicted criminals under sufficient control to be of use). This lead to his featured role during Marvel's Dark Reign storyline circa 2008-2010 or so.
More recently, Phil Urich (who, despite not using the Goblin formula, did have a bout or two with mental illness) found yet another cache of Goblin weapons - this time, one of the Hobgoblin's. Thinking he's killed the original Hobgoblin (actually, the original's brother), he takes on the identity for a short time. Roderick Kingsley, who's been out of the country for several years, comes back and ultimately reclaims the identity of the Hobgoblin. His current modus operandi is leasing the costumes and gadgets of late/retired super-villains out to others for fun and profit.
Phil survived giving up the Hobgoblin identity, becoming the Goblin Knight to the Goblin King (Norman Osborn, again).
NOTE: We'll ignore the "test-tube" "fill-in" Green Goblin Norman used shortly after returning from the dead; Norman's artificially aged children/test-tube babies/whatever with Gwen Stacy (the son eventually used the Goblin formula); and Menace (yet another character who found one of the Goblin's caches and used the formula; she had the advantage of dating Harry (and having an affair with Norman (ick))).
In the 90s cartoon, Hobgoblin came first.
After the Hobgoblin was stopped, the suit was sent to Oscorp where it was originally created. Then when Norman had his gas accident he used the goblin suit as it helped him stay alive due to a certain type of contamination of the suit, and thus becoming The Green Goblin.