Do these sentences imply the same thing?
“I’m going to Europe next week, and then America a week later.”
“I’m going to Europe next week, and then America the week after.”
Also, are they idiomatic?
Do these sentences imply the same thing?
“I’m going to Europe next week, and then America a week later.”
“I’m going to Europe next week, and then America the week after.”
Also, are they idiomatic?
For me these are both idiomatic. I would tend to say "to America" rather than just "America", although informally your version is OK.
The two sentences mean similar things, but the second one is less precise.
For example, suppose you are going to Europe on Monday 6th, and then to America on Thursday 16th. It would be inaccurate to say "a week later", because it's actually 10 days later. However, it would be plausible to say "the week after", because the week including the 16th is the one after the week including the 6th.
"a week later" means 7 days from a starting point
"the week after" means some day during the following week
In practical conversation I wouldn't interpret a strict 7 days for the former but would expect a difference of about 1 day whereas for the latter, I'd interpret as a 7 day window that you could be referring to.
"the week after" could even more-so be perceived as the following calendar week as in "the week after this Friday" could refer to next Monday, Tuesday, Wednesday, Thursday, or Friday where "a week from this Friday" would mean next Friday.