First: according to the established school, the Eid sermon consists of two sermons, whereas the prayer for rain has only one sermon.
Second: in the prayer for rain, the sermon may be delivered either before or after the prayer, whereas in the Eid prayer it is delivered after the prayer.
Third: in the Eid sermon, the rulings pertaining to the two Eids are explained, whereas in the prayer for rain one frequently seeks forgiveness and supplicates for rainfall.
His statement, “opening it with the takbir, as in the Eid sermon”: It has previously been mentioned that, according to the well-known position of the school, the Eid sermon is opened with the takbir, although there is a difference of opinion on the matter. Some scholars maintain that it should be opened with praise of Allah, as the Prophet, peace and blessings be upon him, would do in all of his sermons; and the same applies to the sermon for rain.
Indeed, if someone were to say that the sermon for rain should begin with praise of Allah, unlike the Eid sermon, this would be a sound view, because the Eid sermon occurs at a time when we have been commanded to frequently proclaim the takbir.
His statement, “and frequently seeking forgiveness therein”: seeking forgiveness is to ask for forgiveness, such as by saying, “O Allah, forgive us,” “O Allah, we seek Your forgiveness,” and similar expressions.
Forgiveness means concealing a sin and pardoning it: that Allah conceals the sin and pardons it, so that He does not hold you accountable for it. The word is derived from mighfar, the helmet a fighter wears on his head to protect himself from arrows, lest they strike him.
It is well known that a helmet provides two things: concealment and protection.