Why do Roman Catholics show devotion to the Sacred Heart and Immaculate Heart?

The Sacred Heart of Jesus and the Immaculate Heart of Mary have been intimately related to one another ever since the moment of Incarnation. Most parents know, especially Mothers, that if there is one thing that connects them to their child, it is the heart. Faithful Christians throughout the Church's history have deductively assimilated the connective reality between the hearts of Jesus and Mary through study of the Gospels, private revelation, and reason. Like all orthodox Christian practices, these devotions flow from Christ's divinity.

The Catholic Encyclodepia presents three foundational aspects for the devotion to the Sacred Heart of Jesus: the historical, the theological, and the scientific:

(a) Historical foundations

In approving the devotion to the Sacred Heart, the Church did not trust to the visions of St. Margaret Mary; she made abstraction of these and examined the worship in itself. Margaret Mary's visions could be false, but the devotion would not, on that account, be any less worthy or solid. However, the fact is that the devotion was propagated chiefly under the influence of the movement started at Paray-le-Monial; and prior to her beatification, Margaret Mary's visions were most critically examined by the Church, whose judgment in such cases does not involve her infallibility but implies only a human certainty sufficient to warrant consequent speech and action.

(b) Theological foundations

The Heart of Jesus, like all else that belongs to His Person, is worthy of adoration, but this would not be so if It were considered as isolated from this Person and as having no connection with It. But it not thus that the Heart is considered, and, in his Bull "Auctorem fidei", 1794, Pius VI authoritatively vindicated the devotion in this respect against the calumnies of the Jansenists. The worship, although paid to the Heart of Jesus, extends further than the Heart of flesh, being directed to the love of which this Heart is the living and expressive symbol. On this point the devotion requires no justification, as it is to the Person of Jesus that it is directed; but to the Person as inseparable from His Divinity. Jesus, the living apparition of the goodness of God and of His paternal love, Jesus infinitely loving and amiable, studied in the principal manifestations of His love, is the object of the devotion to the Sacred Heart, as indeed He is the object of the Christian religion. The difficulty lies in the union of the heart and love, in the relation which the devotion supposes between the one and the other. Is not this an error long since discarded? If so, it remains to examine whether the devotion, considered in this respect, is well founded.

(c) Philosophical and scientific foundations

In this respect there has been some uncertainty amongst theologians, not as regards the basis of things, but in the matter of explanations. Sometimes they have spoken as if the heart were the organ of love, but this point has no bearing on the devotion, for which it suffices that the heart be the symbol of love, and that, for the basis of the symbolism, a real connection exist between the heart and the emotions. Now, the symbolism of the heart is a fact and every one feels that in the heart there is a sort of an echo of our sentiments. The physiological study of this resonance may be very interesting, but it is in no wise necessary to the devotion, as its foundation is a fact attested by daily experience, a fact which physiological study confirms and of which it determines the conditions, but which neither supposes this study nor any special acquaintance with its subject.

See http://www.newadvent.org/cathen/07163a.htm

Veneration of the Heart of Mary is analogous to worship of the Sacred Heart of Jesus. It is, however, necessary to indicate a few differences in this analogy, the better to explain the character of Roman Catholic devotion to the Heart of Mary. The Devotion to the Heart of Jesus is especially directed to the "Divine Heart" as overflowing with love for humanity. In the devotion to the Heart of Mary, on the other hand, the attraction is the love of her Heart for Jesus and for God. A second difference is the nature of the devotion itself. In devotion to the Sacred Heart of Jesus, the Roman Catholic venerates in a sense of love responding to love. In devotion to the Heart of Mary, study and imitation hold as important a place as love. Love is more the result than the object of the devotion, the object being rather to love God and Jesus better by uniting one's self to Mary for this purpose and by imitating her virtues

The Catholic enceycopedia also highlights the slight difference in the historical and theological growth of these closely related devotions:

The history of the devotion to the Heart of Mary is connected on many points with that to the Heart of Jesus; nevertheless, it has its own history which, although very simple, is not devoid of interest. The attention of Christians was early attracted by the love and virtues of the Heart of Mary. The Gospel itself invited this attention with exquisite discretion and delicacy. What was first excited was compassion for the Virgin Mother. It was, so to speak, at the foot of the Cross that the Christian heart first made the acquaintance of the Heart of Mary. Simeon's prophecy paved the way and furnished the devotion with one of its favourite formulae and most popular representations: the heart pierced with a sword. But Mary was not merely passive at the foot of the Cross; "she cooperated through charity", as St. Augustine says, "in the work of our redemption".

See http://www.newadvent.org/cathen/07168a.htm