Origin of the names “thionyl” and “sulfuryl”
Different nomenclatures by IUPAC
For the convenience of the reader, here are the names of the compounds as mentioned in the IUPAC's 2005 Red Book[1, p 129]:

$\ce{SO2Cl2 = [SCl2O2]}$ – sulfuryl dichloride or sulfur dichloride dioxide[a], sulfuryl dichloride[b], dichloridodioxidosulfur[c];
$\ce{SOCl2 = [SCl2O]}$ – thionyl dichloride or sulfur dichloride oxide[a], sulfurous dichloride[b], dichloridooxidosulfur[c]

[a] – acceptable common name, [b] – functional replacement name, [c] – systematic (additive) name 
The names which will be pertinent to this discussion are marked with italics, they should otherwise still be upright.
Also note that $\ce{H2SO4 = [SO2(OH)2]}$ is sulfuric acid[a], dihydroxidodioxidosulfur[c]. The molecular entity $\ce{H2SO3 = [SO(OH)2]}$ is sulfurous acid[a], dihydroxidooxidosulfur[c].
TL; DR: The general scheme of things is acid $\to$ radical, and the suffix used in this purpose is -yl. The difference comes from overlap as &quot;properly&quot; both radicals would have the same result: sulfuryl.
Etymology and history
Historically, the terms 'sulfureous' (sulphureous) and 'sulfurous' (sulphurous) date back to the 16th century. Lavosier was first to introduce the adjectives 'sulfuric' and 'sulfurous' to name acids containing sulfur in 1787[2]:

SULPHURIC, SULPHUROUS
The adjectives sulphureous and sulphurous meaning &quot;pertaining to, or of the nature of, sulphur&quot;, date from the 16th century. The well-known oxy-acid was then known as the oil of vitriol (q.v.). In 1787, in the Méthode de Nomenclature Chimique, Lavosier Proposed the name sulphuric acid (Fr. sulfurique) for oil of vitriol and sulphurous acid (Fr. sulfureux) for the related acid containing proportionately less oxygen.[2, p 208]

The suffix -yl designates (hypothetically) as having derived from something.[2]

-YL
The suffix -yl, adapted from Fr. -yle, is derived from Gk. hylē which, in addition to meaning &quot;wood&quot;, also means &quot;the stuff or raw material (of any kind) from which a thing is made.[2, p 231]
[---]
(2) In the name of a radical formed by the removal of a hydroxyl group $\ce{-OH}$ from an acid.[2, p 232]
[---]
(b) Inorganic acids. A number of oxygenated inorganic radicals may be regarded as derived from acids (or from hypothetical acids) by the removal of one or more $\ce{-OH}$ groups. Such a radical is normally named by replacing the suffix -ic of the name ofthe acid by -yl. The more important of these radicals are sulphuryl $\ce{SO2\!\!:}$ (cf. sulphuric acid $\ce{SO2(OH)2}$) ... nitrosyl $\ce{NO}-$ (cf. nitrous acid $\ce{NO$.$\!OH}$)[2, p 232] [---]

As we can see, the -yl name does not care about the oxidation state of the acid. For sulfur, we see that both sulfuric + -yl and + sulfurous + -yl would give the same result: sulfuryl. One of them had to be different. For this purpose, the Greek language was used for the radical of sulfurous acid, and a Latin (possibly Sanskrit[2, p 20]) form remained for the radical of sulfuric acid.[2] This is an historic accident, it could have easily been the other way round.

THIONYL
The name given by Schiff in 1857 to the radical $\ce{SO\!\!:}$ as in thionyl chloride $\ce{SOCl2}$. It was formed from Gk. theion, sulphur, + -yl (q.v.). The radical may be regarded as the characteristic group of sulphurous acid $\ce{SO(OH2)2}$; compare sulphuryl $\ce{SO2\!\!:}$, the characteristic group of sulfuric acid.
(It is fortunate that the L. and Gk. names for sulphur (sulfur and theion) are quite different so that two different names could be formed for the two radicals.)[2, p 214]
SULPHURYL
The radical $\ce{SO2\!\!:}$, the characteristic group of sulphuric acid $\ce{SO2(OH)2}$ (see -yl), e.g. sulphuryl chloride $\ce{SO2Cl2}$ (1867)[2, p 208]

For the functional replacement name, $\ce{SOCl2}$ is named sulfurous dichloride to tie the radical more directly to its acid as well, sulfurous acid. Historically, as noted, the approach here was to use different languages of origin for the main element. This was possible because the words were very different.
If you wish to operate with common names, there is always some historic memorisation. Since this is a binary system, you could remember one and automatically derive the other. But as stated, this will not work for other similar cases: nitrosyl is for nitrous acid radical, not nitric acid.

References and bibliography
[1] Connelly, N. G.; Damhus, T.; Hartshorn, R. M.; Hutton, A. T. Nomenclature of Inorganic Chemistry; The Royal Society of Chemistry, 2005. ('IUPAC Red Book')
[2] Flood, W. E. The Origins of Chemical Names, 1st Edition.; Oldbourne, 1963.