The short answer to this is that Easter is based around Passover, a Jewish holiday, which follows the lunar calendar, while the date of Christmas (introduced later) was based on the Roman Calendar.
Easter, despite its current name being ostensibly Pagan, developed much earlier than Christmas, and was based on the final days of Christ's life, which occurred during Passover (the Last Supper was a Passover Seder). The timing of this date was pretty much entirely taken from Jewish tradition. Andrew McGowan writes:
"We need no quest for the 'origins' of Easter, because at least in one sense they are self-evident; we need rather to describe the processes of reception, interpretation, and transformation that lead from Jewish Pesaḥ to Christian Pascha." (McGowan, 229).
Early Christians saw a theological link between Christ's death and blood and the sacrificial lamb's death and blood that was scattered on doors to protect the Jews from death in Egypt (Exodus 24:8). Christ was seen as the new "Lamb of God," a metaphor now encapsulated in the well-known phrase "Agnes Dei." This description was early enough to be included in New Testament letters as well as the Gospels (cf. John 1:29-36, 1 Peter 1:19, 1 Corinthians 5:7). In Corinthians Christ is actually referred to as the Passover itself. Initially, the remembrance celebration of Easter included only the meal itself. The elaborate Holy Week celebrations describing the final week of Jesus' life did not develop until several centuries after his death (McGowan, 230). This is important to note, because the other celebrations associated with Easter did not come from Jewish tradition. This can sometimes obscure the Jewish roots of the central celebration.
We cannot move on to discussion of Christmas without first discussing even more precisely the date of Easter because even contemporary Christians do not agree on it. While Catholic and Protestant Christians celebrate Easter according to the Gregorian calendar, Eastern Orthodox Christians celebrate Easter according to the Julian calendar. Debate over when Easter should be celebrated is as old as the celebration itself. As Christianity spread, Christian leaders had to contend with different cultural practices and beliefs regarding fasting:
"The Christians were united in believing that they should fast before the celebration, but the different times of breaking the fast undermined the solidarity of the believers." (Stewart-Sykes, 205).
Furthermore, early Christians sought ways to distinguish themselves from Jews. They had already done this by changing their two regular weekly days of fasting to be different from the Jews, but many wanted to set themselves apart from the Jewish celebration by holding it on the Sunday after Passover, the tradition that eventually prevailed.
There is less clarity when it comes to the origins of Christmas. It certainly developed later than Easter, after Christianity had thoroughly separated itself from Judaism, and most likely after Constantine.
A lot has been made of Easter being related to a European Pagan festival celebrated at the same time, and although they may be temporally related, Easter's origins certainly do not lie in European Paganism. Christmas is more ambiguous. Although it was probably celebrated prior to Constantine, it probably wasn't celebrated on December 25th. An 11th century author claiming to be a 4th century Christian writes:
"It was a custom of the pagans to celebrate on the same December 25, the birthday of the sun, at which they kindled lights in token of festivity. Accordingly, when the church authorities perceived that the Christians had a leaning to this festival, they took counsel and resolved that the true Nativity should be solemnized on that day.” (Cohen, 17)
Although the veracity of this quote should not be trusted, what it does tell us is that someone in the 11th century was aware that Christmas had not always been on the day it was then celebrated on, and trying to explain it. Christmas was not always on December 25th. Indeed, for the Eastern Orthodox it still isn't (EDIT: as /u/ThePeasantKingM pointed out below, technically Eastern Orthodox Christians do celebrate Christmas on December 25th; the date labelled December 25th is just different). It was originally celebrated on January 6th. Additionally, the original celebration may have actually been in recognition of Christ's baptism rather than his birth. There were many different groups that had all kinds of mathematical calculations to get themselves to the date of January 6th.
There are basically two theories today, one arguing that the date of birth was calculated from the incarnation which was thought to be around Passover--basically Christmas came, in a roundabout way, from Easter. The second, that Christmas was based on a Pagan holiday celebrating the Sun.
Theory 1: The first reference to Christmas as December 25th comes from 354. An earlier document places Christ's birth on March 28th and claims that this was also the date of Passover when he died. This was part of a common trend at the time--many early suggestions for the date of Jesus' birth attempt to relate it back to Easter/Passover, in order to bring a sort of cosmic complete (circular) narrative to his story. This trend also sometimes referred to the Incarnation rather than the birth, claiming that Christ had been incarnated (conceived, but don't tell theologians I used that word) around March 25th and then calculating 9 months later for his birth to reach December 25th.
Theory 2: So was the date moved to accommodate a Pagan celebration? Most of the evidence from this comes from much later sources, which makes it highly questionable. The author I quoted above was writing in the 11th century, and Jablonski wrote in support of this in the 18th century. It is actually more likely that the reverse is true--a Pagan celebration was invented to piggyback off of popularity of the established celebration of Christmas:
" Julian the Apostate, the nephew of Constantine who sought to restore the “old” ways of pagan religion, seems to be the first to claim (in 362, long after the evidence of the Philocalian calendar) that the December 25 festival was an ancient Roman observance—trying to gain leverage from an established Christmas feast, and inventing a past wherein the divine sun had the prominence he thought it really should have." (McGowan, 255)
The reasoning for the shift may have been theologically motivated. Augustine claims that Christ was born on the winter solstice, so that his life would be parallel to the lengthening of days (and therefore the lengthening of light) (Augustine, Sermon 192).
Regardless, by the time Christmas became an established celebration, it had lost any intrinsic connection to Easter, and thus any intrinsic connection to Passover and the Jewish calendar. Whether it was due to Pagan association or not, Christmas became linked to the winter solstice of the Gregorian calendar rather than any pre-existing celebration based on the lunar calendar. Meanwhile Easter, although it was moved slightly away from Passover in actual date, still retained its connection to the timing of the Jewish feast, which is based on the lunar calendar.
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