LAWN CARE is the process of getting your yard/turf to where it should be.  Basically getting your yard to a healthy state.  This can include things like fertilizer, weed control, disease control, and fungus control.  Lawn Care is typically done periodically as needed.

LAWN MAINTENANCE is doing the routine services such as lawn mowing, trimming, cleanup (leaf removal, etc).  Lawn maintenance is typically done weekly or bi-weekly to ensure your yard is kept healthy and well manicured.