The Liturgy of the Hours is the official prayer of the Church. The Rosary is a private and approved-public devotion. The one is essential to the life of the Church, sanctifying the hours of the day; the other is a haphazard litany that can be prayed by anyone at any time, and unstable by nature. The former is as ancient as the Church; the latter is an invention of the 13th century, fully developed only by the 15th.
If you have time for only one of the two, then for the love of God, stick to the LotH! It has readings from the Fathers, diverse prayers, and the actual Scriptures from which the Rosary meditations are derived.