Difference between prions and amyloid proteins?
Amyloids are protease resistant insoluble fibrils formed because of (mis)folding and aggregation of soluble proteins (Rambaran and Serpell, 2008, Sabate et al., 2015). The first definition of prion was given by Prusiner (1982):


  Because the novel properties of the scrapie agent distinguish it from
  viruses, plasmids, and viroids, a new term "prion" is proposed to
  denote a small proteinaceous infectious particle which is resistant to
  inactivation by most procedures that modify nucleic acids.


Many prionic diseases are characterized by amyloid fibril formation and therefore some (if not many) researchers consider prions to be a subset of amyloids.


  Prions are considered a subclass of amyloids in which protein
  aggregation becomes self-perpetuating and infectious. 
  (Sabate et al., 2015)


However, the amyloid structure formation is not essential either for the autopropagation or disease progression (Moore et al., 2014).


  When wild-type C57BL/10 mice are infected intracranially with the RML
  mouse prion strain, PrPSc accumulates in diffuse nonamyloid deposits
  and brain pathology is distinguished by the hallmark gray matter
  spongiosis for which TSEs are named. Conversely, when transgenic mice
  homozygous for PrPC lacking the GPI anchor (Tg44 or “anchorless” mice)
  are infected with the same RML strain, the incubation time is longer
  and PrPSc accumulates in dense perivascular amyloid plaques with
  little or no gray matter spongiosis.9−11 Similar amyloid PrPSc
  deposition and disease resulting in death has been observed in humans
  with an aberrant stop codon in the C-terminal portion of the PrP
  sequence, resulting in the generation of PrP lacking the GPI anchoring
  group as well as some amino acid residues.13−16 Thus, GPI-anchoring of
  PrPC to the plasma membrane appears to be a primary factor determining
  the type of PrPSc deposition and the development of brain pathology.


You can say that the prion protein PrPC is amyloidogenic but it is not an amyloid. 

There are non-pathogenic (and in fact useful prions) found in fungi. Many of these beneficial prions also form the amyloid (like) structure but they can be disaggregated by chaperones (Wickner et al., 2018). Now again, it is not essential for these beneficial prions to form an amyloid structure.


  Prions (defined as “infectious proteins”) need not be amyloids. The
  vacuolar protease B (Prb1p) of Saccharomyces cerevisiae is made as an
  inactive precursor that is normally cleaved and so activated by the
  action of protease A (Pep4p) [32]. However, in the absence of Pep4p,
  mature active Prb1p can activate its own precursor [33]. This
  self-activation can propagate and pass from cell to cell in an
  infectious manner, acting like a prion, and named [BETA] [34]. In the
  absence of Pep4p, [BETA] is necessary for meiosis and spore formation
  and for survival in stationary phase [34]. It is thus a functional
  prion.




[SMAUG+] is another example of a non-amyloid yeast prion which has beneficial effects (Chakravarthy et al., 2019 and Itakura et al., 2019; Jarosz group). Unlike Prb1p, [SMAUG+] actually forms self-propagating aggregates.

My summary

Many known prions are amyloidogenic and can switch to amyloid form. However, as per the definition, prions are "infectious" or in other words, self-propagating proteins. 
What qualifies as a prion depends on our definition. Mode of action of most known prions is catalyzing a change in the protein fold. Wickner et al (as quoted above) reason that the original definition does not explicitly include the mode of action and hence proteins that activate themselves via proteolysis should also be considered prions. Jarosz's group has shown an example of a non-amyloid prion that forms self-propagating aggregates. It quite likely that more such proteins will be discovered in near future.

In any case, "amyloid" is a structural property whereas "prion" is a functional property and they cannot be considered a sub/superset of one another, even if they happen to be correlated. 

The bottomline is: 
Amyloid is a form of supramolecular protein structure which is often formed by many prionic proteins. However, not all amyloid forming proteins are prionic and not all prions necessarily form amyloids.