From the point of atom efficiency, methylsulfonate is advantageous compared to the p-toluenesulfonate.
On the reagent itself, the methanesulfonate is more polar than the p-TsOH; so the later dissolves better in less polar organic solvents than the former. Methanesulfonate even may be used to dissolve some metal salts.
If used to generate a better leaving group, on workup, a methanesulfonate enters the aqueous phase easier than the tosylate.
p-TsOH is a cheap, widely available solid (sometimes sold as hydrate) you may prepare by your own in large quantities; methanesulfonate however is a liquid, hence potentially easier to add continuously, and more expensive (even more so, per mL, the triflates).
Methyl sulfonates are able to alkylate DNA, hence potentially carcinogen. For tosylates, currently there is not such evidence (but obviously the dust of this strong acid is considered harmful).
As a personal note, among other reasons I found consultation of journals like Organic Process Research & Development just under this aspect (including green chemistry) as useful in two ways: i) publications describing how running a particular reaction within a chain of others works well / better than an other -- especially on larger scales of application--, and ii) the sections in line of "Some Items of Interest to Process R&D Chemists and Engineers".
Addendum: There are dedicated compilations to figure out scope and limitations of chemical reagents. To mention only two, Fieser & Fieser and EROS edited by Leo Paquette (there is an e-EROS as electronic ressource, too) are multivolume references.