Answer:
Great question.
Perhaps the most significant difference is in terms of the Eucharist.
Though positions may vary among contemporary Lutheran denominations today, Martin Luther also taught that Christ is truly present in the Eucharist.
What made his teaching different from the Catholic Church was the philosophical language (metaphysics) he used to describe the Real Presence of Christ.
Luther explained that Christ was present by way of consubstantiation rather than by way of transubstantiation. Both Catholic and Lutheran theologians draw from Scripture to support their explanations, and both also draw upon philosophy to lend support to their explanations. In addition to Scripture and philosophy, Catholic theologians also draw from early Christians and the Church Fathers to help explain that a complete change of substance takes place.
Transubstantiation
In transubstantiation, the bread and wine completely change into the Body, Blood, Soul and Divinity of Christ, with only the outward appearances of bread and wine remaining.
Church Fathers such as St. Ambrose would employ analogies such as the complete change of the Staff of Moses into a serpent and the change of the Egyptian waters into blood to help explain the complete change that takes place at Mass. (cf. Ott 1960, p. 382)
Other Fathers such as St. Cyril of Jerusalem would use vivid analogies such as the wedding feast at Cana to help explain the complete change that takes place at Mass; e.g. Once at Cana in Galilee by a mere nod He changed water into wine, and is it now incredible that He changes wine into blood? (Cat. Myst. 4, 2, cited in Ott 1960, p. 382)
Examples from other Church Fathers can be multiplied. However these examples are sufficient to note that while the word transubstantiation was coined by 12th century theologians such as Roland (later Pope Alexander III), Stephen of Tournai and Petrus Comestor (cf. Ott 1960, p.379) in a philosophical context, it would also be used to shed light on what the earlier Church Fathers said about the Eucharist. This might be likened to using the word Trinity to shed light on Scripture, even though the word Trinity is nowhere to be found in Scripture.
Because the Church found transubstantiation compatible with Scripture, early Christian writings such as the Church Fathers and with philosophy and theology as well, it was later defined as an article of faith required to be held by all Catholics at the Council of Trent (13th session, ch. IV).
Moreover, the Council of Trent (1545-1563) saw other competing philosophical descriptions, such as consubstantiation, as endangering the Scriptural, apostolic and historical understanding concerning the Real Presence. In Her understanding of Herself, The Church has the right and the duty, for the protection of the heritage of Faith, of proscribing philosophic teachings which directly or indirectly endanger dogma. (Ott 1960, p. 9)
Consubstantiation (cf. Sacramental Union)
Martin Luther, on the other hand,
maintained that after the consecration, both the substance of bread and wine together with Christ's Body and Blood are present. He used the term consubstantiation, explaining that Christ is present in the Eucharist in the same way heat is present in a red-hot iron. His ideas about consubstantiation contradict the Church's teaching that the substance of the bread and wine completely change into the Body and Blood, [Soul and Divinity] of Christ, with only the accidents (properties) remaining. (Armenio 2006, p. 472)
Luther hypothesized in 1520 as follows:
Why could not Christ maintain His body within the substance of the bread as truly within its accidents? Iron and fire are two substances which mingle together in red-hot iron in such a way that every part contains both iron and fire. Why cannot the glorified body of Christ be similarly found in every part of the substance of the bread? (Luther, Selections from His Writings, 1962, p. 267-268)
Early descriptions of consubstantiation such as this would later be theologically refined into a concept recognized among Lutheran theologians as sacramental union. It is worthwhile to note that because Luther was influenced by the works of William of Occam (cf. Armenio 2006, p. 471), it is not unreasonable to conclude that he applied the philosophical principle of Occam's Razor (i.e. the simplest explanation is often the best) to support his position for consubstantiation when he writes
Some time ago, when I was studying scholastic theology, I was greatly impressed by Dr. Pierre d'Ailly, cardinal of Cambrai. He discussed the fourth book of the Sententiae very acutely, and said it was far more likely, and required the presupposition of fewer miracles, if one regarded the bread and wine on the altar as real bread and wine, and not their mere accidents - had not the church determined otherwise...I found peace in my conscience in accepting...that the true flesh and the true blood of Christ were in the true bread and true wine, and this not otherwise, nor less, than the Thomists regard them as under the accidents. (Luther, Selections from His Writings, 1962, p. 265)
Foundation for Ecumenical Dialogue
Although Luther was rather polemic against Scholastics such as St. Thomas, he did want to maintain a focus on the Real Presence of Christ in the Eucharist, whether one chose the philosophical aspect of transubstantiation, or consubstantiation as the metaphysical language of description:
I would therefore allow anyone to hold whichever [philosophical] opinion he prefers. The only thing I aim at for the present is to banish scruples of conscience, so that no one may fear being called a heretic if he believes that the bread and wine on the altar are real bread and wine. (Luther, Selections from His Writings, 1962, p. 266)
Thus Luther's focus on the centrality of the Real Presence of Christ over and above philosophical descriptions serves as an ecumenical bridge for deeper dialogue between contemporary Lutherans and Catholics today.
Later followers of Luther however, such as Carlstadt, would take his position to further extremes, denying the Real Presence of Christ in the Eucharist altogether:
on Christmas Day 1521, Carlstadt proceeded to say Mass in German without vestments, publicly denying the real presence of Christ in the Eucharist. Luther would condemn Carlstadt and try to bring about more moderate reforms. Carlstadt, and later his successor Zwingli, would continue to push his ideas further, contributing to the eventual growth of Calvinism. (Armenio 2006, p. 472)
However, even in these more extreme cases such as the denial of the Eucharist being the Real Presence of Christ,
It is a recognized custom for Catholics to meet for frequent recourse to that prayer for the unity of the church with which the Saviour himself on the eve of his death so fervently appealed to his Father: That they may all be one (Jn 17:20).
In certain circumstances, such as in prayer services "for unity" and during ecumenical gatherings, it is allowable, indeed desirable, that Catholics should join in prayer with members of other Christian churches and communities. Such prayers in common are certainly a very effective means of petitioning for the grace of unity, and they are a genuine expression of the ties which still bind Catholics to their separated sisters and brothers. For where two or three are gathered in my name, there am I in the midst of them (Mt 18:20). (Flannery 1996, p. 509)
REFERENCES
Armenio, P. ed. Socias, J. The History of The Church - A Complete Course, The Didache Series, (Woodridge, IL: Midwest Theological Forum, 2006).
Martin Luther. ed. Dillenberger, J. Martin Luther - Selections from His Writings, (New York: NY, Anchor Books, 1962.)
ed. Flannery, A. The Basic Sixteen Documents, Vatican Council II, (Northport, NY: Costello Publishing Company, 1996).
Ott, L. Fundamentals of Catholic Dogma, (Rockford, IL: Tan Books and Publishers, Inc. 1960)