Any grout between the tile and tub should be removed so pull it out with the old sealant and re seal the joint with the most expensive 100% silicone sealant you can find, not just tile and tub caulk. It is necessary to thoroughly remove all of the original silicone in order for the new silicone to stick. Unfortunately that is extremely difficult. Here are some suggestions from GE: "It is VERY difficult to remove silicone from a surface. However, if you must remove it, follow the suggestions below. To remove silicone sealant from surfaces, first remove as much as possible by cutting/peeling/scraping excess sealant from surface. * For ceramic tile, marble, Formica, fiberglass, etc., use 100% mineral spirits (turpentine) and a non-abrasive scouring pad. Test solvent on a hidden area of the surface to ensure that discoloration will not occur. If discoloration does occur, contact the manufacturer of the surface for further assistance. * For glass surfaces, use a razor blade to remove as much as possible, then apply mineral spirits. Remove excess as much as possible, then apply mineral spirits. Remove excess with a towel or other suitable cleaning utensil that will not mar the surface (such as a non-abrasive pad). NOTE: For surfaces such as hard plastics or painted surfaces, including cars, use rubbing alcohol and a soft cloth. Do not use mineral spirits. Only use these solvents in a well-ventilated area and follow all safety precautions and instructions listed on the product label. Material Safety Data Sheets for GE Sealants & Adhesives products are available upon request. Similar information for solvents and other chemicals you choose to use with GE products should be obtained from your suppliers. When solvents are used, proper safety precautions must be observed. On Rough, Porous Surfaces: To remove sealant from a porous/rough surface, (concrete, brick, wallpaper) remove as much of the sealant as possible (same as smooth surface). If necessary, use a wire brush in conjunction with mineral spirits. NOTE: We do not recommend use of a wire brush to remove sealant from wood surfaces, as doing so could damage the wood. Also, mineral spirits should not be used if the wood has any type of finish on it. Test solvent on a hidden area before applying. Mineral spirits are flammable and should be used away from sparks, flames and other sources of ignition. Only use these solvents in a well-ventilated area and follow all safety precautions and instructions listed on the product label. Material Safety Data Sheets for GE Sealants & Adhesives products are available upon request. Similar information for solvents and other chemicals you choose to use with GE products should be obtained from your suppliers. When solvents are used, proper safety precautions must be observed. NOTE: There is nothing that will dissolve silicone. NOTE: If reapplying silicone to the area, remove the old sealant. Then clean the area with a disinfectant if mold or mildew is present, apply rubbing alcohol. Let the area dry before re-applying silicone. Do not use soap to clean surfaces to be sealed because silicone will not adhere to surfaces covered with any soap scum." I suspect this is a far more difficult job than you yet realize.

Always helps to know exactly what you're dealing with. Can you call the installers and ask what specific product they used? Then, I'd suggest getting the carpet professionally cleaned. Knowing what the stuff is will help them to decide what products to use to remove it. I'd be really careful trying to remove it myself because of the risk of making a worse mess than what you have now.

You can ignore any advice you want. You have a change of plane there. It is industry wide specs that call for a silicone joint at all change of plane. Care not what the handyman used or is saying, its obvious you doubt their novice advice and its obvious by the quality of the work they are incompetant. Your big concern now is the proper way in which you remove all residue so Silicone can bond properly and so that you dont leave behind any MOLD prone residues under the new silicone install. So many clowns come here with mold under their poor attempt to correct a change of plane by improperly cleaning out latex based caulks and replace with the proper material. Then later claim their un-mold-abale silicone is producing mold. Its an age old mistake and ive told you exactly how to approach the repair. Now im not looking for likes here or thank you's . But if you speak to any one single tile professional that follows the TCNA/ansi youll see what im saying. The way forward is in your hands. Acetone . Use it. and since you are spanning a larger than industry standard 1/8" joint you need to choose your flash point and type silicone properly theres no less than hundreds of 100% silicone blends/types out there.

For tile, you are really close to 'good enough'. just a bit more scraping. A paint scraper will work better than a putty knife. I wouldn't try machine sanding either. Sandpaper would last about 5 nano seconds--clogged with melted caulk and probably shedding the gooey stuff back on other places on the wall.