Mr. Weasley explicitly explains the purpose and advantages of Portkeys in Chapter Six of Goblet of Fire:
For those who don’t want to Apparate, or can’t, we use Portkeys. They’re objects that are used to transport wizards from one spot to another at a prearranged time. You can do large groups at a time if you need to. There have been two hundred Portkeys placed at strategic points around Britain, and the nearest one to us is up at the top of Stoatshead Hill, so that’s where we’re headed.”
So at the simplest level, people would use portkeys when they can’t or don’t want to apparate. And portkeys can easily transport large groups, and it is easier to arrange portkeys at strategic points than to arrange for qualified wizards to help others Apparate. Earlier in the same Chapter Mr. Weasley alludes to why people can’t or won’t Apparate:
It’s not easy, Apparition, and when it’s not done properly it can lead to nasty complications.
Basically, it is too difficult for many people, and it is also risky. Portkeys have neither of those downsides.
Perhaps most importantly, Apparition is apparently one of the most unpleasant sensations a wizard can experience. In Chapter Four of Half-Blood Prince we get the first description of Apparition:
Harry felt Dumbledore’s arm twist away from him and redoubled his grip; the next thing he knew, everything went black; he was being pressed very hard from all directions; he could not breathe, there were iron bands tightening around his chest; his eyeballs were being forced back into his head; his eardrums were being pushed deeper into his skull and then —
He gulped great lungfuls of cold night air and opened his streaming eyes. He felt as though he had just been forced through a very tight rubber tube. It was a few seconds before he realized that Privet Drive had vanished. He and Dumbledore were now standing in what appeared to be a deserted village square, in the center of which stood an old war memorial and a few benches. His comprehension catching up with his senses, Harry realized that he had just Apparated for the first time in his life.
”Are you all right?” asked Dumbledore, looking down at him solicitously. “The sensation does take some getting used to.”
”I’m fine,” said Harry, rubbing his ears, which felt as though they had left Privet Drive rather reluctantly. “But I think I might prefer brooms. . . .”
Later in the same chapter the unpleasantness is mentioned again:
Braced this time, Harry was ready for the Apparition, but still found it unpleasant.
In Chapter Seventeen when the students are about to learn how to Apparate, Harry tells them that it is unpleasant:
“I dunno,” said Harry. “Maybe it’s better when you do it yourself, I didn’t enjoy it much when Dumbledore took me along for the ride.”
All of them seemed awed, rather than put off, when he told them how uncomfortable it was,
With this in mind it is hardly surprising that people prefer not to Apparate. Indeed, no less a wizard than Dumbledore prefers other forms of travel at times, according to Hagrid in Chapter Twenty-One of Order of the Phoenix:
’Course, this lot don’ get a lot o’ work, it’s mainly jus’ pullin’ the school carriages un- less Dumbledore’s takin’ a long journey an’ don’ want ter Apparate — an’ here’s another couple, look —”
Now, of course, even those who don’t like Apparating may use it at times because of its convenience. Indeed, Harry spends much of Deathly Hallows Apparating. But portkeys match the convenience of Apparating without the terrible discomfort. If a portkey is available it is perfectly understandable that even someone who could Apparate would take it.