"Patron" implies a longer, closer relationship than a mere "Client." Being a "Patron" implies a formal or informal long-term commitment to helping support the artist, either by consistently commissioning new works or offering salaried work like tutoring that affords the artist free time to work on their craft. A patron can also help an artist by offering lodgings and also by allowing the artist into their social circle, making introductions that can help secure new commissions. A "Patron of the Arts" can also be taken to mean a wealthy individual who commissions lots of artwork, even though he is not the single "Patron" of any artist in particular.
A good example I can think of is Donatello's relationship with Cosimo de' Medici. Cosimo commissioned a lot of sculptures from Donatello's worship, but outside of Cosimo's politics and business life they also moved in the same social and intellectual circles. Donatello's business partner, Michelozzo, was so close to Cosimo he even traveled with him to Venice after political upheavals saw Cosimo banished from the city. Possibly guilty for having broken one of the most successful artistic partnerships in Florence, upon his return Cosimo ended up putting up Donatello in a property he owned at a rent rate below market price.
An indication that Donatello's relationship with Cosimo went far beyond "customer-client" and well into the "artist-patron" field is Donatello's reaction when Cosimo drew up plans to demolish his workshop, along with the rest of the block, in order to build a new residence. Donatello, a proud but mercurial character, halfheartedly looked for new lodgings but evidently feeling abandoned by his friend and patron he eventually pounced on a commission in faraway Padua to sculpt an equestrian statue of the recently deceased Venetian war hero, Erasmo da Narni, better known as Gattamelata. Although Donatello would go on to sculpt in Padua, Venice, Siena, and eventually return to Florence, he would never work as consistently or live as comfortably as he did when he was in Cosimo de' Medici's good graces.
Another example of a clear artist-patron relationship is that of Raffaello and Pope Julius II. Raffaello, known in english as Raphael, had his start working out of a small workshop the Central Italian state of Urbino. Fortuitous letters of introduction led him to Florence, where he was eventually introduced to Taddeo Taddei, patriarch of a prosperous family of wool merchants (and historic allies of the Medici in the city council). Taddeo was particularly impressed with the young man, and the two struck up a friendship. In addition to supporting Raphael by commissioning works, Taddeo also introduced Raphael to countless artists present in Florence at the time.
Raphael's career could have stopped there. The Taddei were not amongst the wealthiest great houses of Florence, but their friendship, patronage, and introductions would nonetheless have been enough for him to carve out his own niche accepting lucrative commissions in both Florence and his hometown of Urbino.
However, in 1509 Raphael's life would change forever. Giuliano della Rovere had been elected Pope in 1503, taking the name Julius II. Signaling a desire to assert the Papacy's power following the depraved rule of Alexander VII (better known by his birth name, Rodrigo Borgia) Julius II would go on to embroil the Papacy in the Italian Wars, kicking off the War of the League of Cambrai. Julius would also look to affirm Rome as the cultural center of Italy, and had already brought Michelangelo to Rome in 1505 (where he painted the Sistine Chapel, and later designed the facade St. Peter's Basilica). The Florentine architect Bramante was also hired to restructure the apostolic palace, and was later also involved in the design of the new St. Peter's in addition to other urban projects. Bramante recommended Raphael, whose work he had admired in Florence, to be the decorator of the newly refurbished Papal apartments.
At first Raphael was attached to a team of painters. However, Julius was so impressed with his work in a single room that he soon dismissed all other artists, ordered to have older decorations destroyed (even some notable works by Piero della Francesca) and had Raphael paint "ALL OF THE THINGS!" The Papal Apartments to this day represent Raphael's most astounding work. Subject to the whims of Pope Julius (as his Italian Wars went poorly, he decided that some rooms ought to be decorated with scenes of triumph over adversity) he was still working on the apartments when the Pope died in 1513. His successor, Pope Leo X, quickly reconfirmed Raphael's commission.
Other members of the Papal court would attempt to commission works from Raphael, but the apartments took up such a big chunk of his time, only those closest to the Pope managed to do so. He did spend some time painting portraits of individuals whom he could not refuse, notably of Pope Julius II himself and of the notable functionary Baldassare Castiglione, as well as the Papal ambassador of the Duke of Urbino (his homeland). However, no matter how selective he was in his commissions, his workload soon became unmanageable. Raphael, just over thirty, set up a large workshop in Rome where he and his apprentices pumped out masterpieces at a frenetic pace.
Raphael died in 1520 at the age of 37. Giorgio Vasari, his contemporary, writes that he was exhausted from "Amorous Excesses," however, it's also possible that he was simply exhausted, period. Unlike Michelangelo, who considered himself a sculptor first and foremost and only reluctantly took commissions to paint, Raphael was interested in everything and every art form. Unlike Leonardo da Vinci, who would grow bored with works and leave them unfinished (and really, was an engineer at heart) Raphael instead was wholly devoted to his work. His love of art is clear in his fascination with both the ancient art forms he studied in Rome and the hunger and enthusiasm with which he rushed to Florence to study the artists he idolized. Maybe, had things gone a little differently, he would have surpassed them.