Most of the differences between this world and Earth stem from unequal daylight hours across the world.
Areas will consistently receive less light throughout the year as sunlight headed toward them is blocked by other segments of the planet. You will see cold patches, altered seasons, and a host of other changes because of these.
Here's a list of what I can think of:
Complex scenes in the sky at all times
Depending on how spread out your layers are, you may see the other parts of the planet, in detail, in broad daylight. This is the most obvious change, and it may affect developing cultures. (You could come up with a hundred possible local myths, such as spiders in the sky that spin webs of terrain, or how the sky "cracks" every day and is saved when a god (the moon) appears)
Segments will constantly separate the landscape from the sun, most likely in a predictable way, and native life will get used to it. As a result, timekeeping systems may develop from this (instead of 24 hours per day, 15 eclipses per day - etc).
"Poles" at the joints
Terrain where multiple segments connect will be in shadow for most of the day. Therefore, it may be similar to Earth's poles - colder due to less direct sunlight. This would be be extremely interesting, as it yields a plethora of effects:
Exploration difficulty. Intelligent species seeking to move outward from the segments they inhabit will find that it gets increasingly colder the closer to the joints they move. Harsh temperatures, combined with mountainous terrain due to high gravity and "segment fusion", could even separate populations of people until they developed space faring technology.
Strange winds. Warm(er) air from the centers of the segments will rise, moving toward the joints; cool(er) air from the segments/poles will move toward the centers of the segments. This may be an additional obstacle for travelers, and keep them at home. To add to this, gravity will pull the planet's atmosphere(s) inward, but centrifugal force will pull them outward, potentially causing more strange winds at the segments connecting spheres.
Biodiversity. Intelligent species aren't the only ones affected by segment isolation! Most species will not be able to cross the segment divides, meaning evolution may look completely different all across the planet. Biodiversity is also fueled by surface volume - there's so much space to cover that you're bound to end up with variety.
Coldest at the "true" poles and on the inside
Sunlight will be least direct at the "poles" of the whole structure - the top and bottom - so they will be the coldest regions. Additionally, the inside, while receiving more direct light than the poles, may see much of it obstructed by the surrounding shells - also making it slightly cooler. The converse of this is true - the outer shells will be warmest, and the equator(s) will be warm as well.
Screwy satellites and unsafe orbits
If the majority of would-be impacts just pass through the lattice, and every segment is as thick as the Earth, some objects may orbit specific segments. You may see regions with their own local moons, whose orbits don't stray into other regions. Additionally, comets, asteroids, and other bodies may fling themselves in and out of the planet, or in a complex orbit inside the spheres. Imagine how culture could change as a result of so many objects to see! Consider also that some of these may be their own (spherical planets) within the main one.
A true core
This planet's center of gravity will likely be close to the center of all of the concentric spheres. Some bodies passing in and out of the planet may even lose momentum and settle in this region! Consider a planet or two, or at the very least a gas cloud, to put in this area.
Rich culture overall
This setup has produced regionally divided groups of people, huge biodiversity, potentially large temperature gradients with such a large sphere, and a ton of astronomical phenomena to influence mythology, timekeeping, and more.