There are several differences between צָעִיף and מַסְוֶה.
The first difference to notice is that מַסְוֶה is a noun in the מַקְטֵל form that is based on a verb root ס-ו-י/ה by prefixing the letter מ. Nouns of this form are objects that perform the action indicated by the verb. So, nouns of this form are categorical nouns, in the sense that any one of any number of objects that performs the action of the verb can be referred to with this same noun, although commonly only one particular type of object is intended. Another example of this type of noun that is similar to מַסְוֶה is מִכְסֵה, a covering, as in Genesis 8:13.
In the case of מַסְוֶה, the verb on which the noun is built, סוה, is likely related to an Aramaic word meaning to look, to look like, or to appear as. So the noun means, an object that makes something look like or appear as. In other words, a mask or disguise, not an item of apparel.
The word צָעִיף is a simple type of קָטִיל form noun from the root צ-ע-ף. So it denotes a particular type of object rather than a category of objects.
These two words, צָעִיף and מַסְוֶה only appear a few times in the OT. The word מַסְוֶה appears three times, all in the same context of Exodus 34, in relation to Moses. The word צָעִיף appears three times, in two contexts, Genesis 24 and Genesis 38. In both of these contexts צָעִיף refers to a garment that a woman, Rebeccah or Tamar, uses to cover her head and face.
In Genesis 24:65, צָעִיף is preceded with the article of particularity ה, meaning "she took the shawl". In Genesis 38:14, צָעִיף can be read as either "a shawl", or "the shawl", and in Genesis 38:19 צָעִיף is in possessive form, her shawl. These usages indicate that the צָעִיף is apparently a garment that all women posses and wear in one way or another.
The verbs used with these nouns also differ. For מַסְוֶה, Moses is referred to as "putting it on his face", "removing" it, or "returning it to his face".
For צָעִיף, the verbs are "covered [herself]", "covered [herself]" and "removed".
That is, the מַסְוֶה is referred to as only covering the face, whereas the צָעִיף is referred to as covering the person.
Both מַסְוֶה and צָעִיף are used commonly in post OT Hebrew, to indicate respectively, a ruse or mask and a thin type of shawl commonly worn by women in the Middle East or pre-modern Europe.
Regarding the translation tradition, the targum attributed to Onkelos, which is a literalist translation, uses different expressions for צָעִיף and מַסְוֶה. It uses עיפא for צָעִיף and the contrived expression בית אפי, a "face housing" or "face covering" for מַסְוֶה.
The Jerusalem targum attributed to Jonathan, which is an interpretive translation, also uses two different words for צָעִיף and מַסְוֶה. It uses רְדִידָא, a thin cloth for צָעִיף, and סוּדְרָא, meaning a kind of bandana garment, for מַסְוֶה, that is, it uses the nouns referring to corresponding male and female garments that are commonly used to cover the head.
I think that the word "veil" used in nearly all of the common English translations of Exodus 34:33-35 is intended in the sense of an object that veils - obscures or hides, whereas in Genesis veil is used in the sense of the name of the common female garment. Unfortunately this translation choice leads to confusion. I would resolve this by using the word "covering" for מַסְוֶה, which would both remain closer to the form of the Hebrew noun and differentiate it from the shawl meaning of צָעִיף in Genesis.