LC, further to springhaze's advice, I recently set up an xe account and am awaiting my first transfer to be complete.
I was able to set up the account completely online, by submitting (over a secure server):
1. A scanned copy of the photo page of my passport
2. A scanned copy of my UK bank account statement (within 3 months and with my current address)
3. A pdf file statement from my US bank account (which my bank gives the option of downloading)
I was fortunate to lock in a rate of 1.38 a few weeks ago, so that is one advantage - knowing the rate you will get.
The key differences between the way you described doing it (bank-to-bank), which is what I used to do when I sent funds from the UK to the US many years ago, are the fees and the exchange rates.
Maybe someone else on this board knows for sure, but I *think* that xe gives you a better rate than a bank will, but there are still two types of costs incurred with xe:
1. The xe transaction fee (which is relatively low)
2. The cost of a wire transfer from your bank to the xe bank
Rather than spending money with Western Union for a wire transfer, I opted to have the Bank of America (which I bank with) cut a check and send it to xe, care of their Bank of America branch address in Blaine, Washington, which is their US address. It would have been far easier to simply do a transfer from one BofA account to the other, but for two reasons that wasn't possible (which I won't make this thread any longer with by explaining here!)
ETA: My example is in the context of moving money from the US to the UK, but of course xe can handle other types of transfers!