Different Lectionary Readings For Sundays According to The Episcopal Church, there are (for the most part) two sets of lectionary readings on a given Sunday. Ostensibly, one is for observance of The Daily Office and the other is for The Holy Eucharist.

Why is this so?
The traditional pattern of Anglican worship consists of three main public services: Holy Communion, Matins and Evensong; also called the Eucharist, Morning Prayer and Evening Prayer respectively.  Morning and Evening Prayer are together known as the Daily Office.

One of the great difficulties with choosing a suitable lectionary (i.e. schedule of readings), apart from selecting appropriate passages, is that some people attend church more often than others. 

At the Reformation the Church of England's original  plan was that the psalms were divided into sixty portions and each portion was allocated to be sung either at morning prayer or evening prayer on a particular day of the month (1 to 30).  Most of the rest of the Old Testament was allocated to be read once a year, with a different  Old Testament reading assigned to both Morning Prayer and Evening Prayer for each day of the year. Usually the readings were consecutive, so the reading at evensong would start where the reading  at matins finished, and the reading at matins the next day continue  from where evensong had left off.  The New Testament was assigned to be read three times in the year, again with a portion assigned to morning and evening prayer of each day of the year.  In addition smaller, more select,  passages were allocated  as the epistle and gospel readings for Holy Communion, in most cases following the pre-Reformation pattern. 

So a person who attended both morning prayer and evening prayer ever day of the year would hear or sing the psalms twelve times, and hear most of the Old Testament once and the New Testament three times.

There were complications even then. In February the last few psalms were omitted and in long months the psalms for the 30th were repeated on the 31st. Special psalms (called   proper psalms) were assigned to Christmas Day, Ash Wednesday, Good Friday, Easter Day, Ascension Day and Whit Sunday which broke up the cycle. Proper readings were also assigned to some holy days. In the case of fixed holy days this was no problem as the standard calendar readings took account of these e.g. the old testament lesson on 26th December carried on from the one on the 24th, with  special ones on Christmas Day. In the case of movable feasts this meant that certain readings in the standard calendar  were simply omitted in the years they coincided with a movable feast. 

The major problem though was that not everybody went to church twice a day, 365 days a year, and so for them the plan didn't really work. Someone who attended only evening service, for example,  would generally hear only every other  chapter and not know what had happened in between, like missing an episode of a soap opera today.  Worse, those who only went on Sundays, effectively heard only random chapters. A particular Old Testament passage fell on a Sunday on average only once in 7 years and sometimes it would be 11 years before a passage came round (due to the effect of leap years), and even then it might be omitted if it fell on Easter Day or Whit Sunday. 

It was then decided that, to reflect the fact most people only  went on Sundays  (still to both morning and evening prayer), rather than have them hear effectively random chapters, the most important Old Testament passages should be selected  and assigned as proper lessons for Sundays. This meant that such people would hear every year the most important chapters (or what the authorities believed to be the most important chapters).  It also further disrupted the annual cycle though since  the passages assigned for a particular date would be omitted that year if the date fell on any  Sunday, not just a few movable feasts.

Since that time, both in England and in Anglican churches worldwide, there have been several changes, generally trying to reflect the contemporary churchgoing habits of the majority. It is not possible to suit everybody, from the person who attends one service a year, always  on Christmas Day,  and complains they never hear  what happened when Jesus grew up, to the person who attends every service there is and complains about hearing the same readings again and again.   

In The Episcopal Church  (the Protestant Episcopal Church of the United States, or US branch of the Anglican Church, to be more specific for non-Americans) the most common pattern of  regular Sunday  worship today is to attend one service, Holy Communion i.e. Eucharist. The second most common is to attend a Eucharist and morning or evening prayer. It is less common to attend only morning or evening prayer and not a Eucharist.   For this reason different readings are provided for the Daily Office (morning or evening prayer) because it is assumed that, in most cases, people will already have heard the readings for the Eucharist and not want to hear them again.