These are different words, and their usage (context) differs substantially. How would you define them or explain the difference (if you believe there is one)?
I would say that it is entirely possible to be completely truthful without being at all honest. Generally you do this by answering a question exactly as asked, or by giving as little information as possible when you know something relevant that would interest the other person.
For instance, if your father comes storming into the house and demands, "What happened to the mailbox?!", you could be truthful and say "I saw that on my way in, it looks like somebody clipped it with their car." (After all, you did see it on your way in, and it does look like that). Or, you could be honest and say "My friend who was giving me a ride home wasn't looking when he turned his car around in the driveway and ran into it, and I told him not to worry about it, I'd take care of it for him."
So, for a definition, "being truthful" = "making only statements that you know or believe to be true", while "being honest" = "Telling the whole truth (that is, all the relevant information that you know)."
It looks to me like you're actually asking about the difference between truthful and honest, not truthfully vs. honestly.
As used in conversation and writing, there is very little difference between the two adverbs:
Honestly, I have no idea how that happened.
Truthfully, I have no idea how that happened.
The honestly version has a slight flavor of throwing your hands up in the air and saying "no friggin' clue, sorry"; the truthfully version sounds like you might have spent a bit more time contemplating the situation before reaching your conclusion. Also, as Rosey28 noted, the honestly version is slightly more likely to imply the opposite (i.e. that what you're saying is not actually honest), though truthfully can also be used that way. The only other difference I can come up with is one of usage: usually, you say something truthfully, but admit something honestly. But all of these connotation differences are so slight that it is very easy to come up with counterexamples.
The adjectives, on the other hand, are clearly different. As Hellion noted, it is quite possible to be truthful without being honest. Technically, it's even possible to be honest without being truthful, i.e. if you're mistaken about something. My point is that this difference does not necessarily apply to the adverbial forms, especially when they're used as interjections/conjunctions.
"Truthfully" has a close relation to the issue of fact or fiction. "Honestly" can be used to over-exaggerate one's supposed honesty, and in some contexts, it even suggests a hint of dishonesty.
"I honestly have no idea who hit the mailbox."
Removing "honestly" from the sentence would almost make it seem more truthful.
"I have no idea who hit the mailbox."
In fact, in the legal context, people often add oaths like "honestly" when they are not telling the truth.