Do these two sentences mean the same thing?

tá an ceart ansin agat
tá an ceart agat ansin

If not, what is at play here and are there other phrases which demonstrate the same concept?

English is a funny language - it doesn't seem to tolerate changes in word order as well as our Gaeilge does. The fun we can have!!

Personally, I can't see much difference!

I would usually say
Tá an ceart ansin agat!
..for everything.

Perhaps, just perhaps, if I wanted to make sure to show someone where he was right on paper, I might put it like:
Tá an ceart agat ansin! - and point to the correct place while saying 'ansin'.

But honestly, I think it's down to whatever way you want to put it.
I'd be interested to see if someone could come in and show anywhere where it makes much difference to the meaning.

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Bí cinnte de go nglacfaidh triúr le gach aistriúchán a thabharfar.
Be sure to get three in agreement with a translation given.

Consider English:
Write your mother a letter.
vs
Write a letter to your mother.

The two have the same meaning, essentially, but there is a very, very slight change of emphasis. The first typically places a subtle emphasis on "your mother", making the sentence more about your mother than it is about the letter.

There could be similar effects caused by changing the order of the two in Irish (if it's allowed -- I don't know enough Irish to judge it at all). If I had to make a guess, I'd suggest that perhaps one of them may be a bit... grudging. Sort of like "you're right (this time, but the other stuff you say is wrong).

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A language belongs to its native speakers, and when you speak it, you are a guest in their homes.
If you are not a good guest, you have no right to complain about receiving poor hospitality.

I would agree with Braoin, I don't think there's any real difference in the meaning.