TL;DR: The seasons will have different lentghs depending on the "side" of the horseshoe it is in (there is a slow and a fast "side"), and the day/night cycle will be unchanged.
For more info on horseshoe orbits, there's this Q&A on physics.SE.
From the perspective of an inertial frame in heliocentric coordinates this asteroid [that has a horseshoe orbit] is in a circular orbit (topologically a circle) around the sun.
This means that the seasons will be the same. It has a moment when it will be nearer the sun (summer) and another when it will be farther away (winter). Depending on its rotational tilt, it will probably also have autumn and spring seasons.
But the length of the seasons will be different, and the order may change too. More on that below.
It probably will have rotation, so day / night cycles are a given. Since it is habitable, it is crucial for the cooling of the planet surface to have a day/night cycle. Again, the horseshoe orbit won't affect it for the most of the time.
When it approaches the main planet, things change a bit:
It will eventually catch up with the Earth, but it is not necessarily gravitationally drawn into the Earth. It interacts with the Earth’s gravity field in its frame with an effective and repulsive potential (...). The gravitational potential plus this effective potential pushes the asteroid into a higher orbital radius. The Lagrange points L4 and L5 act then as attractor points in the rotational frame of the asteroid. Its orbital velocity is now smaller and recedes away from the Earth. Eventually the Earth approaches the asteroid and the process is repeated. ref.
The tides will probably become stronger due to their gravity interaction.
Because it will change its orbital velocity as it "rotates" around L4 and L5, the length of the seasons will change also. It will have a faster and a slower season cycle.
Relative to the sun, the planet never changes direction, only speed. On the slow cycle, the main planet catches up from behind and accelerates the horseshoe-orbit planet. It enters the fast cycle, when it goes around the orbit and approaches the main planet from behind, when it has its speed reduced and goes back to the slow phase. All of this while both bodies orbit the sun.
This is a “hunter-chaser” type of orbit. The thinner the horseshoe is the smaller the angular momentum L is with respect to the Earth at close approach. This means the gravitational attraction can become larger. ref