A world with a low axial tilt will have no distinguishable seasons; but would this said world have different biomes across its surface? If so, what would influence the variation of these biomes?
Yes, the worlds would have deserts, rainforests and cooler areas. Without tilt the middle region of the world still has the highest energy density received from the sun, because of the angle. And the north and southpole are still the coldest, because of the angle.
There would still be continental (huge landmasses) and near-ocean-landmass climate differences.
Ocean currents would have a harder time to develop without ice, as they rely on temperature differences. Those still exist on a world without tilt, but are not as huge as on a world with ice.
Without seasons the northpole and southpole are less likely to have ice, but that depends heavily on the average temperature of the planet and the amount of water nearby.
Water helps to average out the temperature as it acts like a huge energy reservoir that takes and gives energy whenever there is a difference to the surrounding air.
A huge landmass at the north or southpole may still be cold enough to have ice, while a small landmass or pure water is less likely to have ice.
The vegetation biome chart on the Wikipedia Biome article is a good illustration of the degree to which climate variation is not based on mere distance from the equator.
Notice that western Europe has a relatively warm, temperate climate, even though it's at similar latitudes as the taiga forests of Russia and Canada. This regional warmth comes from the Gulf Stream that brings warm ocean currents from the Gulf of Mexico to European waters.
Similarly, Peru has a colder ("subtropical") biome for its latitude, which is driven by the Humboldt Current importing cold water from the Antarctic.
Ocean currents are determined by a large number of factors, but the changing of the seasons is a relatively minor one (having the most notable impact on equatorial currents, where the Coriolis effect is weakest).
A planet without seasons will still have a Coriolis force, solar and lunar tides, and wind/ocean currents that are shaped by the contours of land masses. These can drive major changes in biomes independent of seasonal variation.
I mentioned Peru in the context of the Humboldt Current, and I think Peru is a great example case that deserves extra attention. Peru is relatively close to the equator, so you wouldn't expect it to see great seasonal variation the way (for example) northern Europe experiences summers and winters. But check out the different climates in Peru:
Peru has warm and cold deserts, alpine mountains, equatorial rainforests, savannah, and temperate subtropics. The contour of land and the flow of wind and water currents have combined to produce a great variety of biomes within a relatively small area, despite being in an area with reduced (compared to more polar regions) differentiation between summer and winter.
(Note: Peru does experience seasons, but the biome variation across Peru is not created by seasonal variation.)
Assuming the planet is situated at the same distance as Earth from a similar star as our Sun, the climate would be worse than on Earth.
The equator would be warmer than it is on Earth while the polar regions and temperate areas would be colder.
The areas that are temperate on Earth would be permanently frozen. There would be no plant life even at the latitude of Moscow because of no warm season.
The area would be most likely covered with ice, and it will accumulate because of no thawing time.
This will produce chain reaction due to higher ice albedo. The climate of the whole planet would be cooler and most of it would be covered in ice, while a tiny strip on equator would be warmer and dryer.
The poles (especially if there are continents at the poles) would have large areas where sun never shines.