Why does the Eastern Orthodox Church call the 49 days following Easter, Pentecost and the Western Church does not?
To put it in a nutshell: It is their tradition to do so. The Eastern Orthodox Church calls this the Season of Pentecost because there are 50 (Pente) days between Easter and Pentecost. It is that simple.
The Christian holy day of Pentecost, which is celebrated on the seventh Sunday after Easter, commemorates the descent of the Holy Spirit upon the Apostles and other followers of Jesus Christ while they were in Jerusalem celebrating the Feast of Weeks, as described in the Acts of the Apostles (Acts 2:1–31). In Christian tradition, this event represents the birth of the early Church.
In Eastern Christianity, Pentecost can also refer to the entire fifty days of Easter through Pentecost inclusive; hence the book containing the liturgical texts is called the "Pentecostarion". Since its date depends on the date of Easter, Pentecost is a moveable feast.
The holy day is also called "White Sunday" or "Whitsunday", especially in the United Kingdom, where traditionally the next day, Whit Monday, was also a public holiday (now fixed by statute on the last Monday in May). In Germany Pentecost is called "Pfingsten" and often coincides with scholastic holidays and the beginning of many outdoor and springtime activities, such as festivals and organized outdoor activities by youth organizations. The Monday after Pentecost is a legal holiday in many European nations.
In the Eastern Orthodox Church, Pentecost is one of the Orthodox Great Feasts and is considered to be the highest ranking Great Feast of the Lord, second in rank only to Easter. The service is celebrated with an All-night Vigil on the eve of the feast day, and the Divine Liturgy on the day of the feast itself. Orthodox churches are often decorated with greenery and flowers on this feast day, and the celebration is intentionally similar to the Jewish holiday of Shavuot, which celebrates the giving of the Mosaic Law.
The liturgical celebrations of Pentecost in Western churches are as rich and varied as those in the East. The main sign of Pentecost in the West is the color red. It symbolizes joy and the fire of the Holy Spirit. Pentecost (Wikipedia)
It is also the tradition of the Eastern Orthodox Church and those Eastern Catholic Churches to celebrate Mid-Pentecost.
Mid-Pentecost or Midfeast, also Meso-Pentecost (from Greek: Μεσοπεντηκοστή); Russian: Преполове́ние Пятидеся́тницы is a feast day which occurs during the Paschal season in the Eastern Orthodox Church and those Eastern Catholic Churches which follow the Byzantine Rite.
Mid-Pentecost celebrates the midpoint between the Feasts of Pascha (Easter) and Pentecost. Specifically, it falls on the 25th day of Pascha (Easter). At the feast of Mid-Pentecost, a Small Blessing of the Waters is traditionally performed after the liturgy of the feast.
Mid-Pentecost is a one-week feast which begins on the 4th Wednesday of Pascha (Easter), and continues until the following Wednesday. That is to say, it has an Afterfeast of seven days. Throughout these eight days (including the day of the feast) hymns of Mid-Pentecost are joined to those of the Paschal season. Many of the hymns from the first day of the feast are repeated on the Apodosis (leave-taking of the feast). Although it is ranked as a Feast of the Lord and has an Afterfeast, Mid-Pentecost itself is not considered to be one of the Great Feasts of the church year.
The liturgical texts for the feast are found in the Pentecostarion (the liturgical book containing propers for the period from Pascha to Pentecost).
In the West, we call this time frame Eastertide and Paschaltide.
Dom Gueranger has this to say about Eastertide:
The fifty days between Easter and Pentecost have ever been considered by the Church as most holy. The first week, which is more expressly devoted to celebrating our Lord’s Resurrection, is kept up as one continued Feast; but the remainder of the fifty days is also marked with special honours. To say nothing of the joy, which is the characteristic of this period of the year, and of which the Alleluia is the expression,- Christian tradition has assigned to Eastertide two practices, which distinguish it from every other Season. The first is, that fasting is not permitted during the entire interval: it is an extension of the ancient precept of never fasting on a Sunday, and the whole of Eastertide is considered as one long Sunday. This practice, which would seem to have come down from the time of the Apostles, was accepted by the Religious Rules of both East and West, even by the severest. The second consists in not kneeling at the Divine Office, from Easter to Pentecost. The Eastern Churches have faithfully kept up the practice, even to this day. It was observed for many ages by the Western Churches also; but now, it is little more than a remnant. The Latin Church has long since admitted genuflexions in the Mass during Easter time. The few vestiges of the ancient discipline in this regard, which still exist, are not noticed by the faithful, inasmuch as they seldom assist at the Canonical Hours.
Eastertide, then, is like one continued Feast. It is the remark made by Tertullian, in the 3rd century. He is reproaching those Christians who regretted having renounced, by their Baptism, the festivities of the pagan year; and he thus addresses them: "If you love Feasts, you will find plenty among us Christians; not merely Feasts that last only for a day, but such as continue for several days together. The Pagans keep each of their Feasts once in the year; but you have to keep each of yours many times over, for you have the eight days of its celebration. Put all the Feasts of the Gentiles together, and they do not amount to our fifty days of Pentecost." [De Idolatria, cap. xiv.] St. Ambrose speaking on the same subject, says: "If the Jews are not satisfied with the Sabbath of each week, but keep also one which lasts a whole month, and another which lasts a whole year;- how much more ought not we to honour our Lord’s Resurrection? Hence our ancestors have taught us to celebrate the fifty days of Pentecost as a continuation of Easter. They are seven weeks, and the Feast of Pentecost commences the eighth. ... During these fifty days, the Church observes no fast, as neither does she on any Sunday, for it is the day on which our Lord rose: and all these fifty days are like so many Sundays." [In Lucam, lib. viii. cap. xxv.] - THE HISTORY OF PASCHAL TIME
In the West, the Season of Pentecost designates the time after Pentecost until the First Sunday of Advent.