Why Moggallana suffered from past karma effects, but Angulimala did not?

I have heard different classifications of Karma from my Teachers. One such classification goes like the following:

1. Garuka Karma - produces results in the current life or the next for sure. If the karma in question is a bad one, there are five immediate effective karma (pancanantariya karma): killing mother, killing father, killing an Arahant, intentional wounding of Buddha and creation of schism among Sangha. Permanent scepticism (Niyata Micchaditthi) is also regarded as a garuka karma.
2. Asanna Karma - Karma which one does or remembers immediately before the moment of passing away. This kind of karma will play a major part in determining the next birth.
3. Aachinna Karma - Things we habitually perform and/or recollects; for which, one likes the most.
4. Katatta Karma - Any karma that cannot be included in the above categories along with the actions which were soon forgotten are placed in this category.

Remember all classifications apply to both good and bad karma (even Ananthariya karma). Above is not the only classification of Karma.

If you look at the classification, by killing the parents Moggallāna thero had committed a Garuka Karma in previous life which falls to the Ananthariya karma category. It is said to be, that there is no escape from an Ananthariya Karma.

There is no difference.

Both of them "had to face karma", just as all of us have to experience the results of past kamma.

Being arahants, however, they only had to experience the results of the kamma they had created up to the point where they attained arahantship.

Nevertheless, just as we're all conditioned differently based on our individual past kamma, they too were conditioned differently — this is where the difference lies.

In Aṅgulimāla's case, having been a murderer, he had to bear with getting things thrown on him during his almsround — it's just that he didn't die from that particular kammic result. But whenever he did die, though, he did "face kamma" (his very last kammic result) just as Moggallāna would have.

There is some circumstantial evidence that Venerable Aṅgulimāla was deeply hated even after his going-forth into the Dharma.

Then Ven. Angulimala, early in the morning, having put on his robes and carrying his outer robe & bowl, went into Savatthi for alms. Now at that time a clod thrown by one person hit Ven. Angulimala on the body, a stone thrown by another person hit him on the body, and a potsherd thrown by still another person hit him on the body. So Ven. Angulimala — his head broken open and dripping with blood, his bowl broken, and his outer robe ripped to shreds — went to the Blessed One. The Blessed One saw him coming from afar and on seeing him said to him: "Bear with it, brahman! Bear with it! The fruit of the kamma that would have burned you in hell for many years, many hundreds of years, many thousands of years, you are now experiencing in the here-&-now!"

(Aṅgulimālasutta MN 86 as translated by Venerable Ṭhānissaro)

Venerable Maudgalyāyana was murdered by brigands. It is possible that the Buddha would look at that event and also say, like he said of the assault on Ven Aṅgulimāla : "The fruit of the kamma that would have burned [Ven Maudgalyāyana] in hell for many thousands of years, he experienced in the here-&-now!" It's possible. It's also speculation.

Aṅgulimāla and Moggallāna both were Arhants. Aṅgulimāla killed 999. Moggallāna killed his parents in a past life. Only Moggallāna had to face karma?

The act of killing one's own parents was so heinous that it ranks right up there in the same gravity as killing an Arahant and shedding the blood of a Buddha. Refer to the Anantarika-Kamma for further details.