Okay, so I was talking with my old friend John recently about the aftermath of Hurricane Sandy. He commented on how different the weather, and thus human civilization, would be if our planet didn't have a tilted axis. That tilt gives us our seasons and forces changes in the weather throughout the year. Yes, you could also have seasons on a planet with no tilt but with an eccentric orbit which changes the entire planet's distance from the sun. But for now let's just consider some of the ramifications of life on a planet with no axial tilt and a circular orbit.
There is no annual cycle of seasons. There are climatic-zone equivalents of "seasons" based on the latitude of locations: eternal summer in the tropical zone, eternal spring in the subtropics, eternal fall in the higher temperate zones, and eternal winter in the arctic. Other than that there are no identifiable seasons as such.
The calendar would be different. At first the only things you have to mark time are the rising and setting of the sun and the cycle of the moon(s). Once the people on the planet discover astronomy, they can begin to chart the movement in the sky of any other planets in the solar system. But other than astronomical movements, the days and lunar months continue in an unbroken cycle with nothing to mark a "year" in the dramatic and obvious fashion that a cycle of seasons does. Astronomers in pre-modern societies could carve out a prestigious social niche as astrologers, prophets, and oracles based on their knowledge of the heavens.
Dates for religious and cultural holidays would mostly be based on lunar cycles, dawn and sunset, and famous historical (or mythical) events. Lack of seasons means no holidays or festivals based on the solstices
and equinoxes. Also, no welcoming of spring, no mid-summer bonfires, no
fall harvest festival, or other annual observances.
Weather might seem more unpredictable. With seasons you know to expect heat waves in the summer and snows in winter. But without seasons, you have nothing to help predict what, generally, to expect. This might well engender more worship of and sacrifices to deities of nature, particularly the of earth, sea, and sky/weather. If the sky god is propitiated then there will be rains sufficient for the crops, but not for flash floods. People might have more of a craving for stability.
Would there be any animal migrations? Probably few if any. Most are in response to seasonal changes as creature move towards warmer climes in winter, then back as the cold recedes in summer. And what about mating seasons? Mating seasons are typically timed to avoid birthing during winter.
I'm not sure I see a lot of cool campaign potential in such a planet for a fantasy or modern genre game. For a science fiction game it could help make a planet unusual and interesting. Arctic nations with domed cities and economies based on mineral extraction, etc. For a fantasy game I feel that normal seasons--or even exaggerated seasons as in the Game of Thrones stories--engender many interesting events for the players to encounter and more material for GMs to work into stories.
My GMing style involves a good chunk of storytelling designed to present plenty of cultural atmosphere and texture. A big part of culture is expressed in festivals, holy days, and other observances of things important to that culture. The seasons influence all those and also economic activities, such as agriculture and shipping. Descriptions of the gradually changing weather as seasons changes to season help the players feel more connected and immersed.