What do these words mean?

"sarcastic" means saying or writing something in such a way that it makes the opposite point, usually with contempt... For example: looking at a 350-pound man in swimming trunks and saying sarcastically, "There goes a real Olympic swimmer!" or when your child makes a mess in the kitchen and you say, "Well, well, well. I can't WAIT to clean this up!"
"Aesthetic" can mean either the characteristic principles a artist brings to his/her work... such as a fashion designer whose designs always look masculine, or playful, or futuristic... Their tendency toward masculinity, playfulness, or futurism is their "aesthetic".
OR
"aesthetic" also refers to principles of beauty or attractiveness. One plate of food may be artfully and attractively arranged, while another plate of food may be slopped together on the plate. The first is "aesthetically" pleasing, the second is not. This is a rather subjective concept: my sense of aesthetic beauty may be quite different than yours.

Aesthetic- Concerned with beauty or the appreciation of beauty. The pictures, statues, sunsets, etc. give great aesthetic pleasure.

Sarcasm- To mock or convey contempt. Those pants look fantastic on you! While the opposite is the true opinion.

aesthetic = concerned with beauty or the appreciation of beauty. For example: "Those paintings give great aesthetic pleasure." or "I really appreciate the aesthetics of Mongolian sculpture."

sarcastic = marked by or given to using irony in order to mock or convey contempt. For example:
"Would you like me to help you down from that ladder?"
"No, I want to spend the rest of my life up here(!)"
"There's no need to be so sarcastic."

aesthetic is what goes into your arm through a drip when you're going for an operation to put you asleep
sarcastic is when a sardine and a tic can be both cast from a fly rod