What are the main differences between lab-grown tissues and natural tissues from living animals?
Immortalized cell lines are often cancerous, which has large effects on their gene expression profiles. They also adapt to cell culture conditions after several hundred passages in culture. I've cultured both HepG2 cells and primary mouse hepatocytes. The HepG2 are a hepatocellular carcinoma line and grow fine in culture, while the primary hepatocytes don't really grow in culture and only live for a few days after you extract them from a mouse. We were doing gene transfer assays with both sets of cells. The quickly dividing HepG2 cells were easy to transform using PEI, and while primary hepatocytes could be transformed with PEI, they worked better with calcium phosphate, probably because it was less toxic. I noticed that HepG2 cells were smaller than primary hepatocytes, and the primaries were far more fragile, we could run HepG2 through a peristaltic pump with no problems, but had to handle all the primary cells by hand. 

As far as cost goes, tissue culture is almost always cheaper. You can make a bottle of media for a few dollars worth of material and use it for several passages. The cost of running an incubator and biosafety cabinet are also minimal. So unless you're doing something unusual, cell culture is affordable. However, it's an incomplete model at best. Trying to create whole tissues in the lab counts as unusual.

Doing work with animals is far more accurate than cell culture, but much more expensive. Our mice are cheap, but still cost about a dollar each, and about 20 cents a day for housing. A typical biodistribution experiment will use 3 mice per time point and 8 - 10 time points, so 24 to 30 mice assuming no mistakes during dosing. On top of that, doing animal work has a lot more bureaucracy involved, with animal use protocols and approvals and inspections.

The question about whether to use animal models vs cell culture comes down to what you're trying to accomplish. If you're studying gene regulation or protein interactions, cell culture usually works better because it's more amenable to gene transfer and allows you to use techniques like fluorescent fusion proteins and bioconjugation. If you're studying drug metabolism and toxicity you should use whole animals because the interactions between metabolites and specific tissues are hard to reproduce in cell culture. In some far off future we may be able to totally replace animal models with cell culture or even computer models, but we're not even close.