What a fascinating topic! I am a very late-comer to this discussion, just signed up for this site today. There have been a few quality answers already, so just a few thoughts as a biologist with a keen interest in geology and astronomy. Forgive me if I am repeating some other comments.
First of all, I think that the only thing we can say with near-certainty about an earthlike planet without axial tilt, is that there would be no real seasonality, as all others also noted. That's about it, other than that we cannot say things with as much certainty as some others suggest, for the following reasons, let me elaborate a bit further;
Without axial tilt and its resulting strong seasonality, other factors would become (even) much more relevant in determining climate than they are now, in particular:
1) Solar insolation, determined by the type of star and proximity to it. I understand we are here assuming a similar sun and earth combination.
2) Certain large-scale characteristics of the earthlike planet, such as in particular magnetic field and rotational speed (day-length!). Again, I understand we assume earthlike here.
3) Atmospheric composition: we can see in our Earth's past that for instance varying CO2 levels had enormous impact on climates. Higher levels, warmer climates and globally more rainfall; lower levels, colder climates and globally less rainfall.
4) Distribution of land (continents) and sea (oceans), I mean both size and location of continents: obviously very great impact on local solar heating and hence also on winds and rainfall.
5) Mountain ranges: need I elaborate?! Tremendous impact on winds and rainfall.
6) Volcanism: in particular the amount of it, adding CO2, sulphur and many other elements to the atmosphere and biosphere.
7) Life itself: ecosystems, such as large forest areas, have an impact on their own local and even regional climate. 591
For instance, the Amazon rainforest creates part of its own rainfall by a recycling of water.
Summarizing, geographical/spatial differences would become much more determining for climates, by lack of temporal differences, other than day and night. Wind and rainfall would mainly be created by differential (solar) heating as a result of spatial differences, such as land-sea, high-low, north-south, ...
Winds, which are mainly the results of these spatial heating (and resulting air pressure) differences, plus the Earth rotation, would still be blowing, such as the trade winds (though in a somewhat modified form), the equatorial rising airflows, the 'westerlies' and related winds resulting from colder sea / warmer land.
So, yes, there would definitely still be wind, and those bringing rain. We know this, because a lot, if not most wind and rainfall of our present Earth are also caused by relatively short-term differences in heating. But typical seasonal winds, such as monsoons, would be absent.
With regard to life and ecosystems:
What I find remarkable is that the lushest life and the greatest biodiversity on Earth are found in regions with the least seasonality (tropical rainforests and related ecosystems)! Tropical rainforests are climatically defined as places where the day-night temperature differences are greater than the annual/seasonal differences. So, it is absolutely clear that life on Earth does not require an axial tilt and seasons, it is just adapted to it.
At the same time, the equatorial regions also show us how important geographical differences are for creating ecosystems: at the same latitude in Africa and South America we find rainforests, wet and dry savannahs and even semi-arid places.
Therefore, when creating your world, first create the continents and mountain ranges and then look at the major differences in temperature. That should give you an impression of prevailing winds, and rainfall. And with that, give you a start for your ecosystems.
Success! And please let us know some results. 591