What is the difference between a phosphoryl group and a phosphate group?

The difference is basically one oxygen atom. If you consider monoethyl phosphate as an example, the ethyl alpha-carbon bears a phosphate group, whereas the ethoxy oxygen bears a phosphoryl group.

Phosphates are esters of phosophoric acid, $\ce{HO-P(O)(OH)2}$, when one or more its $\ce{OH}$ groups are replaced by (substituded) alkyloxy (alkoxy) groups. E.g. Ethyl phosphate, $\ce{CH3CH2-O-P(O)(OH)2}$. In biochemistry, where the ‘alkyls’ are very complicated, the names are based on the alcohols: ethanol phosphate.

A compendium of various IUPAC books info, IUPAC Gold Book, lacks ‘phosphoryl’ term, however, for ‘phosphorylation’, states:

phosphorylation

A process (catalysed by enzymes) involving the transfer of a phosphate group from a donor to a suitable acceptor; in general, an ester linkage is formed (…)

See, not ‘phosphoryl’, but ‘phosphate’ group.

Phosphoryl is very different group. Cite from IUPAC Nomenclature of organic chemistry, preferred names (2013):

P-67.1.4.1.1.2 Fundamental acyl groups for substituent derived from mononuclear noncarbon oxoacids

(…)

$\ce{-P(O)<}$
phosphoryl

It's the same in inorganic chemistry. E.g. phosphoryl chloride, $\ce{POCl3}$. Cite from IUPAC Nomenclature of Inorganic chemistry, 2005:

Table III, Suffixes and endings

(…)

oryl

Ending of prefixes for substituent groups formed by removing all hydroxy groups from ‘oric’ acids, e.g. phosphoryl, P(O), from phosphoric acid.

I honestly don't know why the process of introducing the phosphate group is called phosphorylation. I guess is sounded much better than e.g. ‘phosphatization’ to the term designer, and analogous to e.g. ‘methylation’, but he/she did not realize, that ‘phosphoryl’ already has a (different) meaning. Btw, the $\ce{-P(O)(OH)2}$ group is ‘phosphono’ group, see e.g. this answer or this question.

UPDATE

… and the (di)anionic form of the phosphono group is named phosphonato group.

P-72.6.1 Prefixes for anionic centers derived from acid characteristic groups

(…)

$\ce{-CO-O^-}$
carboxylato (preselected prefix)

$\ce{-SO2-O^-}$
sulfonato (preselected prefix)

$\ce{-P(O)(O^-)2}$
phosphonato (preselected prefix)

(…)

There's also biochemistry related document^[1], effectively discouraging the use of “phosphoryl” for phosphate group names of biochemicals:

Phosphoric esters, $\ce{RO-PO(OH)2}$, are named as O-substituted phosphoric acids or as substituted alcohols (Table I). Thus, choline O-(dihydrogen phosphate) and O-phosphonocholine are both appropriate names. The latter may be contracted to phosphocholine, but not changed to phosphorylcholine; “phosphoryl” is defined as OP and requires, if used, the naming of all three groups attached to the phosphorus atom. However, “phosphoryl” is retained in derived terms such as the names of enzymes (e.g., phosphorylase) or of processes (e.g., phosphorylation).

(…)

Table I (I won't reproduce it for technical reasons), lists many examples of (1) phosphate names, (2) O-phospho/phospho names, (3) systematic names (the first two types marked as recommended for biochemical usage), and (4) abbreviations. No name uses “phosphoryl”; some examples:

• 1. D-Fructose 1,2-bis(phosphate)
  2. 1,6-Di-O-phosphono-D-fructose;
     1,6-Bis(phospho)-D-fructose
  3. D-Fructofuranose 1,6-bis(dihydrogen phosphate)
  4. Fructose-1,6-P₂
• 1. D-Glyceraldehyde 3-phosphate
  2. 3-O-Phosphono-D-glyceraldehyde;
     3-Phospho-D-glyceraldehyde
  3. 2-Hydroxy-3-oxopropyl dihydrogen phosphate;
     2-Formyl-2-hydroxyethyl dihydrogen phosphate;
     D-Glyceraldehyde 3-(dihydrogen phosphate)
  4. D-Glyceraldehyde-3-P
• (oxidated - carboxylate anionic form of the above aldehyde)
  1. D-Glycerate 3-phosphate
  2. 3-O-Phosphono-D-glycerate;
     3-Phospho-D-glycerate
  3. (R)-[2-Hydroxy-3-(phosphonooxy)propanoate];
     (R)-[2,3-Dihydroxypropanoate 3-(dihydrogenphosphate)]
  4. D-Glycerate-3-P

On the other hand, despite of this recommendation, somewhat related recent document operates with “phosphoryl” name for phosphate unit frivolously.^[2]

If I borrow the image from the neighbor answer

these would be properly names phosphonatooxy and O-phosphonato, respectively. And, for protonated non-anionic forms, phosphonooxy and O-phosphono. No “phosphoryl”.

However it's for names where one needs to name these groups as prefixes. The correct name of the structure in the picture still is ethyl phosphate.

References:

1. Nomenclature of Phosphorus-Containing Compounds of Biochemical Importance. (Recommendations 1976). IUPAC-IUB Commission on Biochemical Nomenclature. Biochem J 1978, 171 (1), 1–19.
2. Blackburn, G. M.; Cherfils, J.; Moss, G. P.; Richards, N. G. J.; Waltho, J. P.; Williams, N. H.; Wittinghofer, A. How to Name Atoms in Phosphates, Polyphosphates, Their Derivatives and Mimics, and Transition State Analogues for Enzyme-Catalysed Phosphoryl Transfer Reactions (IUPAC Recommendations 2016). Pure and Applied Chemistry 2017, 89 (5), 653–675.