They are used loosely as a kind of verbal decoration. At the start of a statement, they both mean "what I am about to say is literally true."
At the start of a statement, "Seriously" means "I am not joking at all."
"Seriously, I think we should not have candy in the house at all, because if there is candy in the house we are both going to eat it."
"Honestly" or "frankly" means "I am not going to soften it the truth, even though speaking plainly might offend you or hurt your feelings."
"Honestly, you need to lose some weight."
As a retort, "Seriously?" means "that's so ridiculous it sounds like a joke." (Usually it is not a joke, and the speaker knows it).
"The politician said that cutting taxes would boost the economy so much that it would actually bring in more tax revenue." "Seriously? Didn't they learn anything the last time they tried that?"
"Honestly!" means the same thing as in a direct statement: "I am about to say something unpleasant, and I am excusing myself for being unpleasant by saying that I am being honest."
"He left his dirty clothing lying on his bedroom floor again."
"Honestly! He's gotten to be a pig. What are we going to do?"