Not a eunuch awn, a unicorn.

If Mythical Creatures Existed no. five: Unicorns

Yes, the dreams of little girls everywhere would be fulfilled. And little girls would be very important in dealing with unicorns, but more on that later.

Points
They can't talk, so that makes things a little difficult. However they are clearly very intelligent animals, and understand a certain amount of what is going on in the human world. India was saved from invasion by Genghis Khan after a unicorn came and knelt before him and he realised invading the country was not the will of God.

Benefits
Well, the horn is clearly the most useful part of the unicorn. It can neutralise poison, which is a very useful thing for people worried about assassination. I also expect it could be put to good use in the medical profession, dealing with snake bites, spider bites, and other poisonous infections. However it would require killing the unicorn just to retrieve one piece of horn, which would not go down well among the various animal rights groups.

Something to consider...
Also, the best way to catch a unicorn is to use a virgin woman to attract it. They are usually wild and strong, but will come and lay their head in a virgin's lap, allowing hunters to encircle it and capture it. I assume there would be an age limit imposed on this practice, to stop little girls being used in contravention of child labour practices.

I also expect Muslim countries will take advantage of this fact to test whether a new bride is in fact untouched. If the unicorn becomes tame, the bride is fine. If not, she gets beheaded (probably).

Natural enemies
Unicorns would need to be kept away from lions, as the two are mortal enemies. And unfortunately for the unicorn, the lion usually wins. This might be surprising to anyone who knows that a single blow from a unicorn's horn can kill an elephant, but I guess lions are a bit more mobile.

Another benefit
If they can be tamed (by virgins, of course), they would make a beautiful addition to our culture. They would ride in parades, and perhaps become the official beast of many royal houses around the globe (cared for exclusively by a sect of chaste women).

Popular culture
I can forsee an issue where women forgo marriage in order to stay caring for their tame unicorns. I have no idea what the effect of marriage would have on the ability of a woman to maintain a good relationship with a unicorn. If it severed their connection, I imagine it would be terrible for both. In that case, I expect to see a number of stories appearing about the woman who had to choose between her man and her unicorn. I also expect to see quite a few teen novels in which young women come of age and leave behind their childhood unicorn friend and move on to an adult relationship with a man (with the unicorn's blessing, of course). These tales would be heart-rending and bittersweet, and would appeal to the over-emotional.

Conclusion
I think unicorns could make a small, but interesting impact on our world. If a captive breeding population could be maintained for the vital medical purposes, then hopefully the world could also enjoy the benefits of having the beautiful creatures in the wild.