They're the same thing - convection ovens are also known as fan-assist ovens (see for example wikipedia), since they're basically an oven with a fan. Maybe what you had previously was marketed as a fan-assisted oven, and threw out the phrase "conventional cooking" to emphasize that you can still do everything you conventionally could. But it sounds like you're just talking about two different convection ovens here.
More likely, the problems you're having are to do with either the convection being more efficient in your new oven, or the temperature control being off on one of the two. Ovens aren't always perfect, so likely your new oven is hotter when you set it to a given temperature than the old one. It might also have a more efficient fan. You can probably learn to adjust by trial and error - reduce temperatures, and check things frequently until you figure out you've gotten it right. It might also be helpful to grab a thermometer and see if your new oven is lying to you about the temperature.
(If you were previously using non-convection recipes in a convection oven, without adjustment, then your old oven was probably either too cool or its fan wasn't doing much. If you were adjusting recipes for convection previously, then maybe your new oven is hotter than it should be.)