What's in a name?

Infertile:

1. Not fertile; unproductive or barren.

2. Biology

   1. Not capable of initiating, sustaining, or supporting reproduction.

   2. Not fertilized and hence incapable of growing and developing: infertile duck eggs.

3. Not capable of sustaining crops or vegetation: infertile land.

4. Unproductive or producing little: a path of research that proved infertile.

Barren:

1. 1. Not producing offspring.

   2. Incapable of producing offspring.

2. Lacking vegetation, especially useful vegetation.

3. Unproductive of results or gains; unprofitable: barren efforts. See Synonyms at futile.

4. Devoid of something specified: writing barren of insight. See Synonyms at empty.

5. Lacking in liveliness or interest.

The American Heritage® Dictionary of the English Language, Fourth Edition
Copyright © 2009

They mean the same thing; barren is, in fact, a synonym for infertile. But I prefer barren. Its old-fashioned sound takes the edge off the bleakness. Plus, good things happen to barren women in the Bible:

¹Now Sarai, Abram’s wife, had borne him no children.... [But God said,] "¹⁶I will bless her, and moreover, I will give you a son by her. I will bless her, and she shall become nations; kings of peoples shall come from her."
Genesis 16:1, 17:16

⁵In the time of Herod king of Judea there was a priest named Zechariah, who belonged to the priestly division of Abijah; his wife Elizabeth was also a descendant of Aaron. ⁶Both of them were upright in the sight of God, observing all the Lord's commandments and regulations blamelessly. ⁷But they had no children, because Elizabeth was barren; and they were both well along in years....¹¹Then an angel of the Lord appeared to him, standing at the right side of the altar of incense. ¹²When Zechariah saw him, he was startled and was gripped with fear. ¹³But the angel said to him: "Do not be afraid, Zechariah; your prayer has been heard. Your wife Elizabeth will bear you a son, and you are to give him the name John. ¹⁴He will be a joy and delight to you, and many will rejoice because of his birth, ¹⁵for ... he will be filled with the Holy Spirit even from birth. ¹⁶Many of the people of Israel will he bring back to the Lord their God. ¹⁷And he will go on before the Lord, in the spirit and power of Elijah, to turn the hearts of the fathers to their children and the disobedient to the wisdom of the righteous—to make ready a people prepared for the Lord."
Luke 1:5-17