Apples and Oranges, bro. The 340 M&P is a pricey gun that delivers superior sights as well as the .357 option; the Bodyguard seems to be geared more for the economy market. Recoil is similar with the same loads, in my subjective opinion.
The CT LaserGrip is much faster to activate than the top-of-the-gun push button on the Bodyguard .38. Of course, it also costs more money.
If you get the new Bodyguard .38 (I'm not talking about the proven old J-frame Bodyguard series that has been street-proven since 1955), be careful when you close the cylinder to index it locked with one chamber in line with the firing pin. On one I've tested, it's possible if the cylinder is closed with the firing pin between chambers and cartridges, to get a "click" instead of a "bang" on the first pull of the trigger. The M&P's trigger pull also seems distinctly better.
You will, however, save a good deal of money on the Bodyguard .38 that comes with InSight laser sight.
Classic example of "ya pays yer money and ya takes yer choice," IMHO.