According to the DMG (239) the proficiency bonus for tool proficiencies don’t apply when making ability checks to ascertain the quality, or to discern the origin, of a crafted items, since these checks do not require tool use.
I strongly disagree with this because no one should be more qualified to appraise and examine an item or structure than someone skilled in its creation.
In the PHB, tool proficiency affords the following:
A tool helps you to do something you couldn’t otherwise do, such as craft or repair an item, forge a document, or pick a lock. Your race, class, background, or feats give you proficiency with certain tools. Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge o f its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver’s tools, or a Strength check to make something out o f particularly hard wood.
I’d expand this slightly, by allowing tool proficiency bonuses on both investigation and appraisal checks made to examine items that would be made with those tools. If the character is proficient in the investigation skill, he does not gain any additional bonus (as he already has a proficiency bonus to such checks).
Furthermore I’d allow characters with a tool proficiency to more capably run a business related to that craft. In the DMG, on page 129, there is a chart that determines how successful a character is in the running of a business enterprise. I’d allow characters with a tool proficiency bonus to multiply that bonus by 5 and add it to the percentile die result. I wouldn’t add any ability score modifier to this bonus because, when running a business, various ability scores would come into play (CHA to deal with suppliers, debtors, and customers, INT to balance the books and deal with the minutia of running a business, WIS to know how to deal with the unexpected, to apply common-sense when tackling problems, and to have the strength of will to weather rough patches, STR and/DEX to work at creating goods for sale, etc.)