How do the rites for a first and second marriage (after death of a spouse) differ in the Orthodox Church?

There are distinct differences in the tone of the two marriage rites of the Orthodox Church. The second is distinctly penitential, with the priest praying specifically for the Lord to have mercy on the couple who have come forward to be married a second time. An excerpt of the prayer following the exchange of rings during the betrothal:

O Master, Lord our God, who sparest all and providest for all, who knowest the secrets of men and hast understanding of all things: Do thou the Fashioner and Creator, who knowest the weakness of human nature, cleanse our sins, and forgive the transgression of thy servants, calling them to repentance, granting them remission of iniquities, cleansing of sins, and forgiveness of transgressions, whether voluntary or involuntary. O thou who didst forgive Rahab the harlot, and didst accept the repentance of the Publican, remember not our sins of ignorance from our youth up. For if thou shouldest mark iniquity, O Lord, Lord who should stand before thee? Or what flesh should be justified before thee?

Contrast that with the prayer immediately following the exchange of rings during the betrothal in the Rite of [a first] Marriage, an excerpt:

For thou, O Lord, hast declared that a pledge should be given and confirmed in all things. By a ring, power was given to Joseph in Egypt; by a ring, Daniel was glorified in the land of Babylon; by a ring, the uprightness of Tamar was revealed; by a ring, our Heavenly Father showed His bounty upon His Son, for He said: Bring the fatted calf and kill it, and let us eat and make merry. By thine own right hand, O Lord, thou didst arm Moses in the Red Sea; by thy true word, the heavens were established and the foundations of the earth were made firm; and the right hands of thy servants also shall be blessed by thy mighty work and by thine upraised arm. And, O Lord our God, do thou now bless this putting on of rings with thy heavenly benediction; and let thine Angel go before them all the days of their life...

There are a few differences that are noted in the text I have linked below for further information, of note is the solemn entrance of the wedding party and that the wedding dress may not be white. Some Orthodox Churches may have their own regulations and protocols, and the presiding priest may even make judgement calls based upon his knowledge of the couples' situation. What's more, some jurisdictions may require counseling and/or meeting with and receiving the explicit blessing of a Hierarch.

In any case, the differences between the two Rites exist because the Church desires to recognize and proclaim the sanctity and holiness inherent in the mystery of matrimony. It is only right to do so because the Orthodox Church believes it to be just that, a Holy Mystery (or Sacrament in the vernacular of the West) that is imminent and occurs just once in the life of the couple. See the OCA's statements on this:

According to the Orthodox teaching, only one marriage can contain the perfect meaning and significance which Christ has given to this reality. Thus, the Orthodox Christian tradition encourages widows and widowers to remain faithful to their spouses who are dead to this world but alive in Christ. The Orthodox tradition also, by the same principle, considers temporary “living together,” casual sexual relations, sexual relations with many different people, sexual relations between members of thie same sex, and the breakdown of marriages in separation and divorce, all as contrary to the human perfection revealed by God in Christ. Through penance, however, and with the sincere confession of sins and the genuine promise of a good life together, the Orthodox Church does have a service of second marriage for those who have not been able to fulfill the ideal conditions of marriage as taught by Christ. It is the practice of the Church as well not to exclude members of second marriages from the sacrament of holy communion if they desire sincerely to be in eucharistic fellowship with God, and if they fulfill all other conditions for participation in the life of the Church.

Since that is the case, the Church moves to draw a distinction between the two rites, the first being joyous and a celebration of the receiving the full outpouring of Christ's blessing and grace into the Mystery, with the second being granted out of Christ's abundant mercy, in the understanding of human weakness and our shortcomings, and that to be able to marry a second time is due to the extension of God's abundant grace and deep compassion for mankind, not man's right or privilege. Thus we see the penitential, almost sorrowful tone weaving through the Second Marriage Rite and climaxing with the joyous prayers of the service of crowning in the end.

See here for the full text of the Marriage Rite and here for the full text of the Rite for a Second Marriage.