What's the name of the 'rubbery' sealant they use between the bath and the wall, and why don't they use it in between the wall tiles? They use a hard grouting between wall tiles, but if the weight of the bath water makes the bath slightly drag the tiles down when it is filled which may crack the grouting (presumably why they use the rubbery sealant between bath and wall), isn't it more sensible to use that rubbery sealant for all the grouting?
The rubbery sealant is bathroom (or general-purpose) silicone caulk. As to why they don't use it between every tile, the answer is that it shouldn't be necessary; except for the one row of tile overhanging the "backsplash" of a "built-in" tub (helps keep minor splashes and overflow from seeping into the wall behind the tub), all tiles should be fixed to the wall, and unless the bathtub is not properly supported from underneath the weight of water should have only a minimal effect on stretching and shrinkage of that wall. If the wall IS stretching and shrinking enough to crack and flake the grout, it will probably also be affecting the tiles' adhesion to the wall, and you will, sooner than you should have to, be paying someone else to repair or replace the tile.
It's also a cost-benefits thing. Grout is very easy to apply; after applying the tiles to the backing, simply load a trowel with the grout mix, and scrape it over tile and gap alike, pressing the grout into the gaps. Then, with a damp rag, wipe off the tile, leaving grout in the gaps. Contrast that with going over every gap between tiles with a caulk gun, carefully applying just the right amount of caulk; caulking is precision work if you want it to look good. You can grout a full shower in less than an hour; you'd be there all day with a caulk gun.
The bath, being made of metal, will expand and contract with temperature changes much more than the tile. So something as rigid as grout will eventually fail. Silicone sealant, or some other flexible bathroom caulk is what is required. Use masking tape on either side of the joint during application to get a professional looking finish.
There're two options - silicone sealant and acrylic sealant, I only tried the first one. silicone sealant is very durable and completely waterproof, has perfect adhesion to most surfaces encountered at bath-to-wall joints. Beware that you have to check the fine print about whether it has adhesion to the specific materials you have at your bathroom.
Those sealants are flexible - if the bath bends slightly the sealant will stretch or compress and the seam remains intact. Still it's nearly impossible to have the bending bath drag the tiles unless tiles are attached to the wall very poorly. At worst the sealant detaches from either the wall or the bath and you have a leakage. It's the mortar-like compound between the tile and the wall that holds the tile in place, not the whatever is between the adjacent tiles.
I really wouldn't use grouting between the wall and bath.
Are you thinking of silicone sealant?
Also when applying use a rubber glove with a bit of washing up liquid on to make it as smooth as possible. Also clean and dry the areas VERY well before applying.