If they are the same weight, then no; nothing distinguishes a .38 lswc from a .357 lswc.
EXCEPT: The alloy and/or casting process used to make the bullet may make it better suited to higher or lower velocity. You mentioned Zero brand bullets, right? Those are swaged lead, not suitable for any load over about 900 fps. Push them faster than that and, soft as they are, they will smear your barrel with nasty streaks of left-behind lead and be inaccurate to boot. However, at normal .38 velocities (whether they are loaded in a .38 or .357 case makes no difference--the velocity you push them at is what matters) the swaged lswc you have will prove excellent bullets.
If you want to load lead bullets at magnum velocities they need to be hard cast, and you may find there are other tricks to employ (precise sizing of your barrel and throats) to make them work without leading. Or, they could have a gas check (a shallow copper cup on the base of the bullet) which prevents hot gasses from chewing up the bullet's hind end.