The IUPAC gold book (IUPAC are the organisation which 'define' standards for things such as chemical nomenclature) describes them as being essentially synonymous. The following definitions are given (link):
Resonance In the context of chemistry, the term refers to the
representation of the electronic structure of a molecular entity in
terms of contributing structures. Resonance among contributing
structures means that the wavefunction is represented by 'mixing' the
wavefunctions of the contributing structures. The concept is the basis
of the quantum mechanical valence bond methods. The resulting
stabilization is linked to the quantum mechanical concept of
'resonance energy'. The term resonance is also used to refer to the
delocalization phenomenon itself.
Mesomerism Essentially synonymous with resonance. The term is particularly associated with the picture of π-electrons as less localized in an actual molecule than in a Lewis formula. The term is intended to imply that the correct representation of a structure is intermediate between two or more Lewis formulae.
There does seem to have been some historic difference that has largely been reconciled into a single definition however one use of Mesomerism that still gets commonly used is the definition of groups as having a positive (+M) or negative (-M) effect to describe the effect of a group on a conjugated system.
The Mesomeric Structures are all the compounds with all possible arrangements or the pi bonds drawn separately. The Resonance Structure is one diagram that uses dashes and lines to denote the partial double bond characters in the compound.
This statement is not entirely correct. The 'Mesomeric' structures you talk about are usually refered to as resonance structures/forms. The depiction with the dashed lines and partial bonds is a resonance hybrid.