In my opinion, the two are inseparable. The main kicker, though, is the fact that religion focuses on the religious institution itself, while spirituality focuses on the change that religious thinking can bring to an individual with regard to their everyday life. I wrote a post here that discusses my thoughts on it.
I, too, come from a Catholic background. We find several of the teachings of Jesus from the gospels that promote a spiritual philosophy:
“Look at the birds of the air, for they neither sow nor reap nor gather into barns; yet your heavenly Father feeds them.” (Mattew 6:26)
“Enter by the narrow gate; for wide is the gate and broad is the way that leads to destruction, and there are many who go in by it. Because narrow is the gate and difficult is the way which leads to life, and there are few who find it.” (Matthew 7:13-14)
“Whosoever will come after me, let him deny himself, and take up his cross, and follow me. For whosoever will save his life shall lose it; but whosoever shall lose his life for my sake and the gospel’s, the same shall save it. For what shall it profit a man, if he shall gain the whole world, and lose his own soul?” (Mark 8:34-36)
“One of the scribes came to Jesus and asked him, ‘Which is the first of all commandments?’ Jesus replied,‘The first is this: Hear, O Israel! The Lord our God is Lord alone! You shall love the Lord your God with all your heart, with all your soul, with all your mind, and with all your strength. The second is like: You shall love your neighbor as yourself. There is no other commandment greater than these.’” (Mark 12:28-34)
“A new commandment I give unto you, That ye love one another; as I have loved you, that ye also love one another. By this shall all men know that ye are my disciples, if ye have love one to another.” ( John 13:34-35)
And of course we have the Sermon on the Mount as the penultimate summary of Jesus’ teachings and numerous parables, my personal favorites being the Good Samaritan, the Prodigal Son, and the Pharisee and the Publican. But we also have a train of thought that exists outside of the New Testament canon as well:
Jesus said, “If your leaders say to you, ‘Look, the (Father’s) kingdom is in the sky,’ then the birds of the sky will precede you. If they say to you, ‘It is in the sea,’ then the fish will precede you. Rather, the (Father’s) kingdom is within you and it is outside you. When you know yourselves, then you will be known, and you will understand that you are children of the living Father. But if you do not know yourselves, then you live in poverty, and you are the poverty.” - Gospel of Thomas, saying 3
Jesus said, “I am the light that is over all things. I am All: from me all came forth, and to me all attained. Split a piece of wood; I am there. Lift up the stone, and you will find me there.” - Gospel of Thomas, saying 77
“His disciples said to him, ‘When will the kingdom come?’ [Jesus said,] ‘It will not come by watching for it. It will not be said, ‘Look, here!’ or ‘Look, there!’ Rather, the Father’s kingdom is spread out upon the earth, and people don’t see it.’” - Gospel of Thomas, saying 113
I think it is worth exploring the connection between Jesus and Greek philosophy. The Socratic tradition was well ingrained in Roman Judea. I tend to think many devout Christians view Jesus almost as a made-up character in a story and don’t even consider the possibility of him engaging with Greek thought. Burton Mack wrote a book titled The Lost Gospel: The Book of Q and Christian Origins that examines the connection between the teachings of Jesus and Cynicism (and I mean the ancient philosophy, not the modern-day idea of being a pessimist). John Dominic Crossan’s book The Historical Jesus: The Life of a Mediterranean Jewish Peasant also explores this relationship. While I think that they diverge from traditional Christian theology, which I uphold, they are excellent reads.
Speaking of institutionalized religion, we have several fantastic writers to read: Church Fathers like Justin Martyr, Athanasius, Maximus the Confessor, Augustine, and Isaac the Syrian, Medieval thinkers like Psuedo-Dionysius, John Scotus Eriugena, Meister Eckhart (who I would call the real “treasure hidden in the field”), the continuers of his thought like Henry Suso, John Tauler, and The Theologia Germanica, and the theology of Gregory Palamas in the Byzantine world, Reformation era thinkers like Johann Gerhard, Jacob Boehme, the Pietists and the Quakers, Spainish writers like John of the Cross, and French writers like Br. Lawrence. Classic modern examples like Thomas Merton are a good read, too. In my opinion, thinking like theirs is a rebellion against strict dogmatism and puts a focus on how the individual puts the ideas of the Bible into practice.
Of course, spirituality isn’t limited to the Bible or Christianity; Confucius, Lao Tzu, Zhuangzi, and Buddha happen to be some of my favorites to read. And Socrates, Plato, Aristotle, and their followers are as spiritual as spiritual can get.