A reaction intermediate is nothing more than a plain old intermediate, which you probably understand. There is no stipulation as to how stable the intermediate must be. The IUPAC Gold Book defines intermediate as:
A molecular entity with a lifetime appreciably longer than a molecular vibration (corresponding to a local potential energy minimum of depth greater than $RT$) that is formed (directly or indirectly) from the reactants and reacts further to give (either directly or indirectly) the products of a chemical reaction; also the corresponding chemical species.
I am not aware of a corresponding "official" definition for a reactive intermediate, but in my experience it simply refers to, well, intermediates that display exceptionally high reactivity. Typically these correspond to high-energy species such as carbocations, radicals, carbanions, or carbenes, which exist only transiently, and are difficult or impossible to isolate. A simple example is the carbocation intermediate in an SN1 reaction.
Obviously all intermediates are reactive to one degree or another – which is why they are intermediates and not simply thermodynamic products – but the term reactive intermediate is usually reserved for those which are particularly reactive. So, reactive intermediates are a subset of reaction intermediates.
As for an example of an intermediate which is not particularly reactive, consider the Staudinger reduction of an azide. The initial steps leading to loss of nitrogen gas are extremely fast and (very vigorous) effervescence is observed immediately upon addition of PPh3:
but the iminophosphorane, i.e. RN=PPh3, reacts with water only very slowly to produce triphenylphosphine oxide and the primary amine. Typically you need to heat the reaction mixture for a number of hours, and if you didn't add water it would probably be quite happy on its own. I would therefore consider the iminophosphorane to be an intermediate (or reaction intermediate if you prefer), but I would not refer to it as a reactive intermediate.