To get a visa for your spouse, you need to submit Form I-130. After processing she will receive a CR-1 (or SR-1) immigrant visa.
The K-3 is optional if you are a US citizen. If you file for a K-3, you still file the I-130 first, then submit a Form I-129F for the K-3. After processing, she will receive a K-3 non-immigrant visa (unless the CR-1 gets processed first).
The big advantage of the K-3 was a shorter wait time before she could travel here, typically 6-9 months vs 9-12 months for the CR-1 immigrant visa. With recent changes in processing procedures, the time advantage is not always in play. Applying for a K-3 doesn’t necessarily stop the CR-1 processing. For some time a while back, spouses were getting the CR-1 approved while CIS was still diddling with the K-3 paperwork.
The visa processing overseas is almost identical.
The CR-1 requires a little bit more paperwork up front but less paperwork overall.
The initial fees for the K-3 are a little bit lower than for the CR-1, but with recent fee hikes I think the K-3 is quite a bit more costly overall.
With the CR-1, she arrives as a legal resident and her Green Card and SS card are mailed to her automatically. With the K-3, she has to (a) apply for adjustment of status after arrival or (b) wait for the I-130 processing and return home to get a CR-1 visa. Most all recipients of the K-3 choose to stay here and adjust status.
With the CR-1, since she has an interim Green Card on arrival at the airport, she can work right away as soon as she has her SS Number (2 weeks?). With a K-3, she must first apply for a work permit (60-90 days) or wait for her adjustment of status (6mos – 2 years?) before she can apply for a SS Card and work.
Depending on your state of residence, it may also be faster to get a driver’s license with a CR-1.
With a CR-1, she can apply for naturalization a little before she has been here for 3 years. With the K-3, she will have to wait for her adjustment of status first, then count 3 years before applying.
If she has children coming with her, as in cowboy’s case, it gets a little more complicated but the advantages/disadvantages are basically the same.
If you have previously filed for numerous K-1 fiancée petitions, have a criminal background, or beat up your previous wife, you may want to go for the CR-1 so you can skip the IMBRA requirements.
Personally, all things considered, I would probably choose to skip the K-3 and wait a few more months for the CR-1.
Ray