Hi,
SPTrustedIdentityTokenIssuer represents an external token issuer trusted by SharePoint, for example, when you configure SAML-based federation authentication with ADFS, the ADFS server is the trusted identity token issuer. You can also understand it as the
trusted identity provider. When user authenticate against SharePoint, SharePoint redirects users authentication to the trusted identity token issuer, and the token issuer assigns token to user to enable user accessing SharePoint.
For more information,refer to the following article:
https://technet.microsoft.com/en-us/library/hh305235.aspx
SPClaimProvider is function to add claims to users token. For example, when a user has successfully authenticated in SharePoint with Windows authentication, the claim provider Active Directory will add security groups which the user belongs to as claims
into the users token. There might be several different claim providers for adding different claims to users token.
More explanation about claim provider is available here:
https://msdn.microsoft.com/en-us/library/office/ee535894.aspx.
Best Regards,
Lisa Chen