Actually, Selmo, the Catholic Church has NOT done away with Purgatory. You are conflating Limbo and Purgatory. Limbo was the theological concept of a place outside of Heaven for those who die without Baptism, yet who lived lives worthy enough of reward, and since Heaven requires Baptism for entry (or so holds Catholic dogma), then there needed, in fairness, to be somewhere for those such as innocent babies or righteous heathens to go after death, since they clearly had not merited Hell.
The current thought in the Church officially does away with that, and replaces it with the dogma of "baptism of desire", the idea that desiring baptism is itself merit enough to go to Heaven. Presumably, this desire is assessed in ways we can't exactly measure.
Purgatory, on the other hand, while certainly less emphasized since Vatican II as a result of an attempt to get away from the whole idea of "Catholic guilt" and get closer to being a people of joy, remains very much a part of Catholic theology.
It is, as has been noted already on this thread, a place for those who have not died in a state of perfect grace, but who are not overall bad enough for Hell. It is important to distinguish between Catholic and Protestant concepts of how Grace works here. Classical Protestantism believes in the doctrine of "sola gratia"-- through grace alone. This is, in fact, one of the great dividing lines theologically between Catholics and Protestants. Catholics hold that Faith AND Works are necessary to get into Heaven. It is not enough just to believe, if you truly believe, you must DO.
Now, it is true that Catholics believe, as Protestants do, that entry into Heaven is impossible without God's grace. However, it is necessary to do your part as well. It's not just a matter of saying "I accept Jesus as my lord and saviour", and POOF!, you're in the lineup for Heaven. In Catholic dogma, you are now eligible... but you still have to deal with those sins, since no one ever stops sinning. Confession (the Sacrament of Penance) and the Eucharist are both ways of erasing sin, but it is almost impossible to be going to confession so often that, when you die, you haven't a sin on your soul.
To draw this back to Tolkien somewhat... let's use the Elves as a comparison. It has already been said on this thread that Mandos seems an appropriate analogy for Purgatory, and I think it apt.
Now, when an Elf dies in good graces with the Valar and with the world in general, it is inevitable that he or she will be reborn, correct? Since it is the inherent nature of the Elves to be reborn, rebirth for an Elf could be equated to Heaven for Catholics. However, Elves are not necessarily reborn immediately after death. While a certain waiting period seems to be in order, the period also seems to vary. Some, such as Glorfindel or Finrod, who would be the Elven equivalents of the Saints, pass through remarkably quickly, whereas the average time in Mandos seems to be somewhat longer, just as most of humanity spends a bit more time in Purgatory than Mother Theresa or John Paul II. Then there are those who take FOREVER... the Sons of Fëanor for example seem to be consigned to particularly long sentences.
I would say, though, that Mandos also functions as "Hell" for the Elves. Hell, by a more modern definition, is a separation from God's presence and love, the natural intended state of humanity. Well, the nature state of the Elves is corporal life in Arda while it lasts, so the state of being permanently separated from it would, by definition, be Hell.
Getting back to Purgatory, it is certainly understood as a place of perfection. Niggle is obviously a good enough man that deserves Heaven (and for Niggle, Heaven is getting to finish his tree, which makes one feel that the phrase "Heaven on Earth" is applicable, and also makes one wonder what tastes of "Heaven" in our lives we will encounter in their fullness on the other side...), but he is not a perfect man...
Now, because the Catholic concept of justification, being made right before God, does not just involve accepting God's grace, but involves work. It is not an instantaneous process, but a more labourious one, and it requires the work of the person it concerns. Because of free will, we can't just be flicked from bad to good. It has to be our effort, though it is only the grace and mercy of God that we have the strength or the guidance to succeed. For Niggle, before he was ready for Heaven, he had various problems with his personality, his management of time and his efficiency being the "allegories" of sinful defects. When his time in his cottage, painting away, which can be taken as his mortal life, is over and he is not yet perfected, he is taken to the workshop to finish the job.
I'm fairly sure I've rambled more than need be... and I'm quite sure there's some repetition with what's already in this thread, and I'm positive more can be said, but I'm at what feels like a good place to stop. Convalescents have the right to be selfish, don't they?
Obviously, as resident Catholic, I can't avoid a thread touching directly on such things... but I am exhausted, and I can only hope the above makes sense. My leg throbs and my butt is numb is my excuse if it doesn't...