I think there are more advantages to getting a dive package than just price. You mentioned your wife is newly certified. Based on your personal info, it appears that you don't have a lot of experience yourself. When my wife and I were beginning, we always seemed to spend the first couple days just getting somewhat comfortable with our skills, with the gear, with the divemasters and with the environment on the boat and in the water. If you go with different dive operators, that's going to change for you, daily, and it will be more difficult to find that relaxed comfort zone that really lets you enjoy the whole dive experience.
I'm assuming you'll be renting gear -- at least BCs and regs. Having a BC and reg that you can get used to and stay with for the week will help you that comfort zone much sooner. Similarly, operators like Sunset House and Divetech require all divers to use computers. (If you have a dive package with them, the computer is included.) I'm not sure what a dive operator would do if you showed up, say two or three days into your trip, and you didn't have your own computer to show your recent dive log and remaining non-decompression time. You'd probably have to switch to using tables, if that's allowed by the operator.
Getting used to the boat, the procedures and the dive staff is another benefit of going with one operator, one package. Each boat functions a little differently. But, more importantly, diving with the same staff can help you a lot. You get to know them and they get to know you. They'll quickly learn what to let you do on your own, what things to keep an eye on and when to get involved. If it's a very good staff, like Divetech, they'll make use of what they see in you and they'll help you improve your skills and comfort level. A DM that knows you're there for more than just one dive will also take more time to show you goodies and critters and will make more effort to help you enjoy your experience.
If shore diving in the afternoons and evenings appeals to you, then definitely go with a package. Again, there's the cost saving issue and more. As you dive with the group, you'll see who you want to dive with, and you'll find dive buddies for the shore dives. Not to mention the possibility of discovering and making new friendships that will extend beyond the week in Cayman.
