What core features distinguish generative phonology from metrical phonology?
Generative phonology is a broad meta-theoretical framework, which posits a phonological component with computational devices for deriving input-output relations. It is a mentalist theory, which purports to model properties of the mind regarding the phonological component, and it relies on the notion of linguistic universals, which are the innate principles that constitute the language faculty. Metrical phonology is a sub-theory of a particular model of generative phonology, the Nonlinear model (which also includes Autosegmental phonology). Sometimes that model is called Autosegmental and Metrical phonology. The previous model of phonology, the SPE model of Chomsky & Halle 1968, is also a model of what a generative phonology is, and is technically fairly different.
The SPE model makes various specific claims about rules and representations, and AMP makes different claims. The SPE model posits a rule theory with a very rich set of notational devices for collapsing simple rules into rule schemata, introducing many mathematical devices such as value variables and string variables. Representations are ordered sequences of segments and boundaries, where the latter are unordered sets of value-attribute pairs drawn from a fixed set of attributes ("coronal", "anterior") and values ("+","-" and integers), where every attribute is present. Rules that delete or insert segments of change the values of particular attributes (features).
Autosegmental phonology eliminates the rich set of mathematical expressions from the rule component, and add sub-grouping within a segment, that is, it orders the features so that voiced, spread glottis, constricted glottis are ordered together and can be identified as a specific subgroup. The theory also relaxes the requirement that all features be present in all segments, or that features have to be in a "segment". The theory of rule formalization is much more restricted, generally being restricted to insertion or deletion of an association relationship or feature node, but the theory of universal grammar is enriched by innumerable proposal such as the Obligatory Contour Principle.
Metrical theory is a sub-theory in the Nonlinear model specifically designed to address speech rhythm and stress. There are many varieties of metrical theory, but the primary division is between tree theory and grid theory (in fact, the original tree theory which is analogous to syntactic trees is no longer practiced). Grid theory posits a relation of "prominence" notated with a star, and some number of layers. The difference between "latest" and "latex" in English would typically be that the first syllable of both words has two layers of star-notated prominence, and they differ in whether the second syllable has one (lowest) layer of prominence (latex) or none.