Pulling the table below from TechNet, the difference between Standard and Enterprise search in SharePoint 2013 is really only:

• Content Search Web Part
• Custom Entity Extraction
• Extensible Content Processing
• Advanced actions on Query Rules
• Video Search

The differences from Foundation to Standard are too many to list separately so I’ll let the table do the talking.

Search features	SharePoint Foundation	SharePoint Server 2013—Standard Edition	SharePoint Server 2013—Enterprise Edition
Advanced Content Processing	Yes	Yes	Yes
Content Search Web Part	No	No	Yes
Continuous crawl	Yes	Yes	Yes
Custom entity extraction	No	No	Yes
Deep links	No	Yes	Yes
Event-based relevancy	No	Yes	Yes
Expertise Search	Yes	Yes	Yes
Extensible content processing	No	No	Yes
Graphical refiners	No	Yes	Yes
Hybrid search	Yes	Yes	Yes
Managed navigation	No	Yes	Yes
On-premises search index	N/A	N/A	N/A
Phonetic name matching	Yes	Yes	Yes
Query rules—Add promoted results	No	Yes	Yes
Query rules—advanced actions	No	No	Yes
Query spelling correction	Yes	Yes	Yes
Query suggestions	No	Yes	Yes
Query throttling	No	Yes	Yes
Quick preview	Yes	Yes	Yes
Recommendations	No	Yes	Yes
Refiners	Yes	Yes	Yes
RESTful Query API/Query OM	Yes	Yes	Yes
Result sources	Yes	Yes	Yes
Search connector framework	No	No	No
Search results sorting	Yes	Yes	Yes
Search vertical: “Conversations”	No	Yes	Yes
Search vertical: “People”	No	Yes	Yes
Search vertical: “Video”	No	No	Yes
Tunable Relevancy	No	No	No