The transmon is a Josephson junction and capacitor in parallel.
Originally, transmons were differential circuits, i.e. two transmons on the same chip were not galvanically connected in any way.
In other words, transmons didn't share a ground reference.
Furthermore, in the early days, transmons were almost always embedded into the middle of a harmonic resonator.
The resonator, often referred to as a "bus resonator", was used to couple multiple qubits together, i.e. qubits embedded in the same resonator could couple to each other.
The important differences with the xmon were that
The xmon was grounded. Each xmon on a chip connects to a common ground plane with a nominally fixed voltage.
The xmon was not embedded into a resonator. Instead of coupling through a resonator, each xmon couples through a direct capacitance to each of its neighbors.
Nowadays, several research groups build qubits without the bus resonator and call them "transmons".
Much more could be written. If someone leaves a comment asking for more details on any particular aspect of the difference between transmon and xmon, I will write more.
History of the name
Rob Schoelkopf told me the story of where the name "transmon" came from while we were at the Les Houches summer school on "Quantum Machines".
The charge qubit suffered from low frequency noisey charge fluctuations that lead to dephasing.
To get around the problem, Professor Schoelkopf thought to shunt the junction with a bit of transmission line.
The line would be a short circuit at dc, allowing low frequency charge to equalize, but it would be a high impedance at the qubit's resonance frequency allowing the resonance to remain.
The combination of a transmission line with the junction plasmon mode lead to the name "transmon".
In the end, it turned out that a capacitor was simpler than a transmission line and served a purpose equivalent to the transmission line, so the qubit wound up being a capacitor in parallel with the junction.
However, the name "transmon" had already stuck (or maybe "capmon" just didn't sound as good).