degene = the one
diegene = that one
The above is the literal difference. But the two expressions are used somewhat differently in English than in Dutch.
In Dutch there is little difference between the two. In English, "the one" is used much more often than "that one", even in contexts where Dutch might literally say diegene = that one.
That resource you linked makes it look like degene would be correct but not diegene: We use degene when talking about a specific person:
Kan degene die mij midden in de nacht gebeld heeft mij voortaan overdag bellen? Could that person who called me in the middle of the night from now on call me at daytime?
Degene die het eerst aankomt, wint een prijs. He who (whoever) finishes first, wins a prize.
Consider the following:
1. He is the one I need
2. He is the one whom I need
3. He is the one who I need
4. He is the one that I need
5. He is the one which I need
For most English speakers, 5 would not be valid, because in contemporary English which is used to refer to things but not persons. Many English speakers would also not allow 4, because they feel that that too should not be used to refer to persons. 3 would be accepted by most English speakers today, who acknowledge (some wistfully) that whom is no longer always required in the objective case. 2 and 1 would both be accepted by almost all English speakers.
The great advantage of 1 is that it avoids both the who/whom and the person/thing issues.
Bear in mind that Dutch does not allow the construction that English uses in 1. In Dutch, a relative pronoun is needed to introduce the subordinate clause.
Royston, if the Owl did not accept your sentence, perhaps it was because it feels that that should not be used with persons. Or maybe it was just sloppy programming.
No. The relative pronoun "wie" is used to refer to people (rather than things), but only in conjunction with a preposition.
So there are several different forms of the relative pronoun, used as follows:
1. Direct reference to a singular person: De man die ik gisteren heb geholpen is nu ziek.
2. Direct reference to plural people: De mannen die ik gistern heb geholpen zijn nu ziek.
3. Direct reference to a singular neuter gender thing: Het boek dat ik nu lees is lang.
4. Direct reference to plural neuter gender things: De boeken die ik nu lees zijn lang.
5. Reference together with a preposition to a singular person: De man met wie ik stond te praten is mijn oom.
6. Reference together with a preposition to plural people: De mannen met wie ik stond te praten zijn mijn ooms.
7. Reference together with a preposition to a singular thing: De stoel waarop jij zit is van mij. Het bed waarop jij zit is van mij.
8. Reference together with a preposition to plural things: De stoelen waarop jij zit zijn van mij. De bedden waarop jij zit zijn van mij.