SRID stands for Spatial Reference Identifier and SRS ID stands for Spatial Reference System Identifier. A spatial reference system defines a projection, coordinate system, datums and ellipsoids. Morten Nielsen gives and excellent definition of What is a spatial reference system?.
The term SRID is usually used in common GIS database lingo as an integer in an OGC compliant database which is the primary key in a lookup table called spatial_ref_sys (or similarly named table or view). It is used to lookup the spatial reference details of a spatial reference system and most OGC SFQL functions simply use this SRID key defined in the table when doing projection based functions.
In other GIS usage the spatial reference identifier is usually called SRS ID (Spatial Reference System Identifier) and it is usually composed of Author Name and the unique Integer ID defined by that authority of that spatial reference system. For example the common WGS 84 long lat has an identifier EPSG:4326. Massachusetts State Plane NAD 83 meters would be referred to as EPSG:26986. In general the same integer ID part is rarely used by two different authorities so then simply saying 4326 people know what that means and a lot of database systems simply use the Authority's Integer ID for their SRID. Also in general most commonly used SRIDs have author European Petroleum Survey Goup (EPSG) so its safe to leave out the author. So in short the terms SRS ID and SRID are often used interchangeably although they have slightly different meaning.
One thing that is particular confusing is that since the database use of the term SRID is simply a pseudo-dummy id in a lookup table and even you can define your own if you are not satisfied with any of the common ones, you can conceivably have two different SRIDs that define equivalent spatial reference systems or map to the same SRS ID. The database will commonly treat them as different since they have different SRIDs and its less taxing to assume they are different. There isn't even an official rule that states your database SRIDs have to match up with official SRS IDs.