Equity is the union for actors and stage managers in the professional theater world. And because they have a weekly minimum wage built into their contracts, when the producers of a Broadway tour have hit most of their major markets and are looking to squeeze the last dime out of their show by cutting costs, they stop using Equity performers. The tour will usually go on hiatus for a brief period and a non-Equity cast will take over. The sets will also sometimes be altered in order to be less expensive.
Non-Equity tours have a bad reputation because they are usually not as high in quality as an Equity tour because non-Equity actors, typically, are only in their twenties (often they are fresh out of college), and are still in the early stages of mastering their craft. It is for this reason that non-Equity tours generally (from my experience), have the quality of a well-done college production. The problem people have with this is that producers will usually promote these tours as the "Broadway production," when many of the performers lack the polish of Broadway actors, the sets are cheap, and very often it has been re-directed by someone other than the original Broadway director. Non-Equity "Broadway" tours are almost always a falsely advertised product. This upsets theater patrons who shell out Broadway prices for something that is not really a Broadway show, and it ticks off Actors' Equity (and Equity actors) to no end for the obvious reaons.
That said, there are a lot of extremely talented non-Equity actors out there who aren't members of the union because they haven't gotten their big break yet, or, as is often the case in "regional" theater cities like Chicago (where I live), because they don't want to have to limit themselves in terms of what roles they can accept (if you are a member of Actors' Equity, you are not allowed to perform unless under an Equity contract). I myself am a non-Equity actor, and even though I am a member of the Equity Membership Candidate program, I have no burning desire to join Equity at the moment.
By the way, I have been in a non-Equity tour. But I am not ashamed to admit it because it was not falsely advertising itself as a Broadway show
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