The best way is to start with this post is to start with an example
1)DRAWING-The basic structure of cartoons are played with to give them amusing looks while in anime/manga characters are well defined and more human like unless they are not aliens or monsters but yet all characters have an element of influence unlike cartoons. <as seen above>
2)EMOTIONS-As people say eyes are important to depict emotions that's what anime does make the eyes bigger,let them have more space on face and it works they do depict emotions well.While in cartoons emotions come through action.They jump in excitement,walk head down in sadness and depression etc etc
3)PLOT- Overall anime are known to have a specific story lines for the targeted audience.They have a clearly defined story line making them more mature as compared to cartoons.While cartoons are lighthearted no specific character development just for fun animations.
The two chases
4)QUALITY-The anime develops as it moves on,making it highly interesting.The quality is impressive with fight sequences generally stealing the show while cartoons are still with hardly any story development.
5)IMPACT-The anime has a much more impact on the audience.They tend to create emotions,you feel the characters making it more lively.Naruto,Fairy tail gives the lesson to care for friends,never give up and enjoy life .Death Note was where you give a possibility of what you can do with a death note.While Cartoons can be said to be made to amuse us,just for gags.
Maturity is what makes anime differ and better than cartoons but sometime we need amusement too :)
PS-Exceptions always arise.