Why is there such an emphasis on Jesus's suffering - particularly by the Catholic Church?

I would respectfully disagree with @bruised_reed's statement ..

Christ's suffering was not unique in respect to the levels or duration of physical or emotional pain

Indeed, it was unique. All of God's righteous wrath was poured out on Christ. A wrath for all the sin that had been committed and the sin yet to be committed. I do not think as finite beings that we can understand just how much punishment was leveled upon Christ. He certainly did, however, and I think we can see a glimpse of the torment awaiting him as he asked Abba, Father, all things are possible unto thee; take away this cup from me: in the garden of Gethsemane. This is Christ, the Son of God, asking, "if there was some other way". To me it shows that Christ knew what was coming and if it was enough to terrify him I cannot even imagine how horrible it was.

I know what the OP means as I used to wonder, "Christ knew he was going to rise again from the dead. He was going to live again. How bad could it have been?" But, I think it was far worse than anything we could ever imagine.

Here's a quote from G.K. Chesterton which I think explicates a Catholic view of the subject:

“That a good man may have his back to the wall is no more than we knew already, but that God could have His back to the wall is a boast for all insurgents forever. Christianity is the only religion on earth that has felt that omnipotence made God incomplete. Christianity alone felt that God, to be wholly God, must have been a rebel as well as a king. Alone of all creeds, Christianity has added courage to the virtues of the Creator. For the only courage worth calling courage must necessarily mean that the soul passes a breaking point -- and does not break. In this indeed I approach a matter more dark and awful than it is easy to discuss; and I apologize in advance if any of my phrases fall wrong or seem irreverent touching a matter which the greatest saints and thinkers have justly feared to approach. But in the terrific tale of the Passion there is a distinct emotional suggestion that the author of all things (in some unthinkable way) went not only through agony, but through doubt. It is written, 'Thou shalt not tempt the Lord thy God.' No; but the Lord thy God may tempt Himself; and it seems as if this was what happened in Gethsemane. In a garden Satan tempted man: and in a garden God tempted God. He passed in some superhuman manner through our human horror of pessimism. When the world shook and the sun was wiped out of heaven, it was not at the crucifixion, but at the cry from the cross: the cry which confessed that God was forsaken of God. And now let the revolutionists choose a creed from all the creeds and a god from all the gods of the world, carefully weighing all the gods of inevitable recurrence and of unalterable power. They will not find another god who has himself been in revolt. Nay (the matter grows too difficult for human speech), but let the atheists themselves choose a god. They will find only one divinity who ever uttered their isolation; only one religion in which God seemed for an instant to be an atheist.”

Orthodoxy

The extraordinary thing about Jesus' suffering is not so much its physical intensity, but the fact that He, the Second Person of the Trinity, deigned to suffer. He did not have to, but he specifically wished to save us by enduring suffering.

We should also keep in mind that Jesus, being God, had (and has) perfect knowledge of all the sins committed in the past, present, and future, and His suffering was made infinitely worse by that knowledge. It was this moral suffering, it seems, that caused Him to sweat blood during the Agony in the Garden:

And being in an agony he prayed more earnestly; and his sweat became like great drops of blood falling down to the ground (Lk 22:34).

Finally, by suffering to expiate our sins, He also entered into perfect solidarity with those who suffer in the world, giving them hope (if they live it united with Him, at least), and a meaning to their suffering.

EDIT: St. Thomas Aquinas supports the view that Jesus' suffering was the greatest both physically and morally. I quote below a portion of his responsum to Summa Theologiae, IIIa Pars, q. 46, a. 6:

[T]here was true and sensible pain in the suffering Christ, which is caused by something hurtful to the body: also, there was internal pain, which is caused from the apprehension of something hurtful, and this is termed “sadness.” And in Christ each of these was the greatest in this present life.

...

The magnitude of His suffering may be considered, secondly, from the susceptibility of the sufferer as to both soul and body. For His body was endowed with a most perfect constitution, since it was fashioned miraculously by the operation of the Holy Ghost.... And, consequently, Christ's sense of touch, the sensitiveness of which is the reason for our feeling pain, was most acute. His soul likewise, from its interior powers, apprehended most vehemently all the causes of sadness.

In other words, because Jesus was perfect man as well as perfect God, he was actually more sensitive to physical and moral sufferings than we would be.

It is possible to hold different opinions about this topic (as long as uniqueness of the act of Redemption is maintained), but Thomas' position is well worth considering.

I think I know what you're getting at with this question, but you've got to consider what St. Paul says:

Col 1:24

Now I rejoice in my sufferings for your sake, and in my flesh I am filling up what is lacking in the afflictions of Christ on behalf of his body, which is the church,

And that was at the beginning of the Church. It's not that St. Paul thought there was anything lacking in Sanctifying Grace, but there are still souls to be saved. 1900 years later St. Faustina repeated the words she heard Jesus tell her in the Chaplet of Divine Mercy:

For the sake of His sorrowful passion have mercy on us and on the whole world.

So one can't take Jesus' suffering out of His sacrifice and leave it as the same sacrifice. Whatever sacrifice Our Lord made, it was done in suffering. Therefore, it is impossible to over-magnify the suffering that undid death and brought us into eternal life. It's simply the best thing and as a Christian you'd do well to think of it often, or do what St. Paul does and rejoice in it!

The Catholic Church doesn't say we need to dwell on Christ's sufferings all the time. Consider the Rosary as a good example of how to divide up meditations on particular aspects of the Life of Christ. Only two days out of seven each week in ordinary time are spent meditating on the Sorrowful Mysteries (The Agony in the Garden, the Scourging at the Pillar, The Crowing with Thorns, the Carrying of the Cross and The Crucifixion).

Fridays as a special day of penance, is a good time to spend over-magnifying Christ's Sufferings. Think also of the intention behind the Holy Martyrs witness. Didn't they consider their sufferings as less important than Jesus'. Many of them died deaths that were possibly more painful:

• Skinned like St. Bartholomew
• Roasted alive like St. Lawrence
• Crushed under a door like St. Margaret Citherow
• Starved and left in a box for weeks like St. Maximillan Kolbe

Many others were crucified as well from Sts. Peter and Andrew all the way up to St. Paul Miki less than 200 years ago. The martyrs are our witnesses and examples of how to live and die for Christ. Not everyone is called to martyrdom, even St. Francis of Assisi had found it too difficult to get himself martryed. Still, he was given the grace to bear the wounds of Christ late in his life.

So, when John says:

John 3:16

For God so loved the world that He gave his world that He gave His only Son

as St. John Paul II points out in Salvifci Doloris you can't separate the word gave from the Christ's suffering and the object of the sentence is

...so that man should not perish but have eternal life

which means, God giving His Son to us is Jesus' sacrifice. Which is why every other evil that is inflicted on the world from acts of violence to what insurance policies call "acts of God" are impotent in their ability to harm an immortal soul.

On the surface, Christ's suffering may not appear necessarily unique in respect to the levels or duration of either physical or emotional pain, but it was certainly unique in respect to the nature of the sufferer and the spiritual dimension of the suffering: The completely innocent Incarnate Creator suffering at the hands of his rebellious creation; and the bearing of the burden of the sins of the whole world and consequential estrangement (albeit temporarily) from the intimate fellowship (enjoyed from eternity past) of the Heavenly Father. (cf. Philippians 2:5-8, Isaiah 53, Matthew 26:36-42 & 27:46)

edit: I believe the preceding paragraph would be largely agreeable to most Nicaean Christians. Catholics in particular are often encouraged to meditate on the suffering of Christ as it helps to develop a greater appreciation of Christ's work overall and leads the believer to turn their heart whole-heartedly to the Lord. An example of this is the contemplation of the stations of the cross.