Great question, compassion and pity are quite commonly misunderstood. And this is equally relevant to both Buddhists and non-Buddhists alike.
While entire schools of Buddhism are anchored on the concept of compassion, and many compendia have been written on the topic, here's an easy way to spot the difference:
Compassion is the ability to recognise the suffering in self or others, coupled with the desire to alleviate that suffering. It's borne in the understanding that everyone is just like you: they are alive and have a strong desire to be happy and to be free of unhappiness. Its foundation is based on the understanding that sincerely, and unselfishly, caring for others is an empowering source of peace and happiness. One only needs to engage in genuinely compassionate thoughts, speech, and actions to experience the truth in that teaching. When we genuinely have concern for the well-being of others and take actions accordingly, we experience a deep sense of positive emotions -- it feels good; it feels open and vast... and that's because it's supposed to, it's a reflection of a heart that is open and vast and a mind that is free of self-cherishing thoughts and pettiness.
Compassion is often coupled with compassionate speech and/or action; genuine compassion is rarely passive. In many schools of Buddhism, we diligently work to cultivate insights into how compassion for others is an inexhaustible source of inner-peace, and though regular insights and practice, compassion begins to flow freely and easily, colouring everything we think, say, and do. It's a wonderful way to live life and interact with the world around us!
Pity, on the other hand, is borne of a better-off-than mindset, and one where help is often rendered with an unspoken expectation of something in return (e.g., appreciation, gratitude, respect, influence, reputation, ego, leverage, reciprocation, etc). It's more passive than compassion and is more often coupled with words-alone or superficial offerings rather than compassionate action. Pity is a reflection of an unhealthy mind and those who engage in acts of Pity, even if only occasionally, can serve as excellent objects for our own compassion.
We must be ever-so mindful of the intentions and motivations behind our actions for others. As such, we must be vigilant when we are on the cushion of the potential dark side of meditation: indifference, detachment, pity, routine, self-cherishing, coldness, and pride; these create a type of blind hardness and can be a challenge to both novice and experienced meditators and dharma practitioners alike. This is where having a skilled meditation trainer and dharma coach to personally help you stay the course becomes important.
I hope this was helpful to you.
Recommended listening: The Science of Compassion: A Modern Approach for Cultivating Empathy, Love, and Connection by Kelly McGonigal
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The Sagacious Buddhist Blog
Michael Turner is a deeply accomplished and realised enlightenment trainer and pre-monastic dharma life coach. He emphasises and teaches the practical application of Buddhism in our everyday lives to make real progress toward Enlightenment and is particularly adept at explaining them in ways that can be easily understood and practiced by Western Buddhists. He has been meditating and cultivating the techniques to generate indestructible happiness and inner-strength for more than 25 years and has helped countless numbers of people enhance their capacity for resilience and happiness in real life.