Two possible answers here:

1) Taxonomies are perhaps more stringent with clear-cut boundaries whereas typologies are often more concept-based

2) Taxonomies often contain several sub-categories whereas typologies often do not (perhaps because they are concept based to begin with)

Usually taxonomies have clearer distinctions and distinctions at higher levels followed by ones in lower levels, with the lower levels often having further sub-categories. A good example would be biological taxonomy. All organisms are broken into one of 5/6 Kingdoms. Within each kingdom, there are further sub-categories known as Phyla. In each phylum, there are further sub-categories (Classes), etc. all the way down to Species. Furthermore, the distinction is usually based on something clear cut; e.g., the organisms' cells have cell walls or they don't. The organism has a backbone or it doesn't.

Typology is usually breaking things into groups without further sub-categories. Often times these things are more conceptual so there is some ambiguity between the types and not quite as clear of a distinction. I am a linguist, and in lingusitics we often use "typologies" because a language is thought to reside in one of several categories, without there being further sub-categories. Furthermore, the rules for inclusion are not always 100% followed, so there are generally examples of a language not following in some cases and instead we aim for what "generally" happens in the langauge. For example, you can have one of 6 different types of language depending on the order of subject, verb and object (e.g., SVO, SOV, VSO, VOS, OSV, OVS). There aren't really many sub-categories of that discussed, although there is some room for argument as to which category a language lands in (e.g., German tends to be SVO in many cases, but in the spoken past-tense the main verb comes at the end, and it shares many qualities that are generally found in SOV languages, so it is not as clearly SVO as some other SVO langauges such as Thai). There are also often exceptions despite the somewhat clear rules. For example, although English is generally an SVO language, we do have older English sentences that violate this (e.g., Down through the chimney came old Saint Nick). However, they are not the norm of the langauge, so English is considered SVO. Notice it is not as clear cut as "has a cell wall" or "has a backbone". Alternatively, when discussing the evolution of languages, the term language families is often used because there might potentially be sub-categories there.

This is just a linguists' perspective, but these two things seem to be the major differenes to me. If anyone else has a perspective from their field, feel free to chime in.

A typology groups concepts by shared traits based on theory, while a taxonomy arranges them hierarchically using systematic relationships.

Hello, in my experience I've learnt that "typology" refers to "type", whereas "taxonomy" is a list, or classification of something. So you can study the "types" of plants and taxonomise them, for example.

Typology has qualitative nature, taking care of quintessence.

Taxonomy uses a quantitative (mathematical, statistical) methods.