History and International Relations student here:
One of the many things that humanities do is teach you how to communicate. I can gather dozens of sources of information together and organise them into a coherent argument. I can argue a point of view verbally in a seminar, i can read through and take in pages and pages of information, being rationally selective of what information i need and what i don't need. I can write eloquently when needs be, i can write persuasively using rhetorical techniques, and i can apply such techniques when talking to other people.
I remember writing a personal statement to back up my application for university. I handed in the first completed draft of my statement and my tutor was very impressed. My other friends, whom were mostly science students, were made to re-draft and re-draft again their personal statements because their tutors were not happy with them. When i got to university one of my history lectures told us that a history degree is synonymous with a communications degree- and i think he's right.
But among other things, humanities helps you to think critically and analytically. Not only when it comes to examining someone's argument and looking for flaws in it that you can exploit, but also in how you become increasingly self-critical of your own work and ensuring that your own argument is solidly backed up by evidence. This process of critical thinking, especially in subjects like philosophy, engages the brain in some highly complex cognitive thought.
Science is largely about knowing facts- how to work formulas, how to do x y and z in a lab, what various particles do and the functions of cells etc. Now, from what i know science students may be sometimes expected to argue a case, but the greater emphasis is however just about knowledge. In humanities, you've got to do both to equal measure. You may, for instance, have to know what the socio-economic affects of the Black Death were, but also argue wherever you agree or not with the notion that these affects are related to, say, the Reformation. The knowledge that science teaches it's students is indeed however more useful. Knowing how certain particles behave in cold air or underwater say is useful to many people whom want to manufacture products. However, whilst the knowledge humanities teaches may have fewer practical applications, we have to do more with that knowledge than science students.
Finally, as has already been hinted at. I think there is some truth behind the idea that studying humanities does improve yourself as a person. For me, studying history has given me a vivid sense of identity and place and the processes in the past which have shaped today. I see things today, like the names of towns on roadsides, or the stars in the sky above, and i can "see" things in my minds eye that only people in the past believed. Personally, i find it very fulfilling. At 6th form college, along with history, i also studied philosophy (which i minored in at university) and English literature. I know, roughly, how to think like a philosopher and this gives me a deep insight about myself and life in general. Studying English Literature, and learning about how writers such as Wordsworth, Shakespeare and Hardy expressed themselves helps me in turn express my own creativity. Creative writing is something i enjoy doing in my spare time, and studying literature had aided me in doing this.
Studying humanities has taught me how to communicate, how to think critically, analytically and martial large amounts of information into a structured argument. It can also be immensely fulfilling, it's (metaphorical) food for the soul. Studying science has many practical applications, and is hence useful on it's own merits. I wouldn't like to compare to the two because they do very different (but often related and complimentary) things. One seeks to understand the world, the other seeks to understand humanity. Any society needs both arts and sciences in order to function with any meaning.