MEME is performing de novo motif discovery. It is considering only the sequence data you provide, and looking for short, similar, sequences that occur more often than would be expected by chance. The short, similar sequences are used to construct a position-specific weight matrix (PWM) that characterizes the motif. MEME reports the PWM, and also reports the best matches to that PWM in your input sequences
AME is testing your sequence data for motif enrichment, that is, do your sequences contain more or better matches to the motifs then would be expected by chance. You provide the motifs to be matched as input. A few matches to a motif may be enough to establish enrichment, but not enough to support de novo 'discovery'.