The answer is actually the same as any other basis for being hired. Say you got hired because of your research in a trendy field, what is your long term status if the field stops being trendy? Possibly not good, but it depends.
Hiring and long term employment are different. In the first case, you are evaluating a stranger based on indirect evidence, in the second case you are able to draw on your experience of working with them. Once you get hired as "X's spouse", you have the opportunity to use the resources of your position and accrue your own accomplishments. This would secure your long term professional status independently of your spouse. In this case you would not be treated differently.
Conversely, if you've been working for several years and the most interesting fact about your career is still that you're X's spouse, then it's a different story. Obviously your independent status is not great - arguably worse, because now you've had an opportunity and made little use of it. Although just as your spouse was once reason enough to hire you, they may now be reason enough to continue employing you or even promote to a more permanent position.
If Ann's job offer is tenure-track (or already tenured), will Bob's be, as well?
It may or may not. This depends on a combination of how desirable a candidate Ann is, how (un)desirable a candidate Bob is, and how Ann and Bob negotiate the offers. Ann could always ask that Bob's offer be made tenure-track as well (and the hiring committee could always say no).
Assuming both jobs are tenure-track, will Bob's case for tenure depend at all on Ann's (and vice-versa)?
Officially it would not, and for obvious reasons, the committee will probably not say "you did okay but let's see how Ann does, if she fails we'll deny yours too". They could possibly be biased in favor of Bob out of concern that Ann will leave if he is denied tenure, or they may be less likely to approve him before Ann becomes tenured than after. At this point it depends entirely on the personal feelings of individual people on the Ann/Bob situation (and institutional policy, if any).
If Ann and Bob get divorced, or Ann leaves for another university, will Bob's job be affected?
Officially it would not, and I doubt they would fire him "because" he got divorced, but they may fire him and give some other reason. More commonly, I think what would happen is that Bob would continue in his present position. The consequences would be more obvious if Bob attempts to advance (tenure review) or if funding deteriorates and the department must choose who to let go. Then it will come down to what Bob has achieved since being hired.
Is the fact that Bob was a spousal hire in any way visible to the general public, in his official job description or resume? What about unofficially?
I have never heard of any official "spousal hire" title, and I doubt it would be spelled out in the contract, as it would accomplish little besides opening up the institution to various lawsuits.
There may however be rules about relationships between employees in general, so for example Bob is not allowed to be Ann's postdoc, so HR may need to be notified about the marriage (not necessarily about whether the hire was connected with the marriage).
Unofficially, of course the hiring committee is aware of it, and people talk (even when not supposed to). Academics are no exception and have their share of gossips. Besides it's not like it's hard to tell when two people are married.
In a completely unofficial capacity, does being a spousal hire in any way hurt the academic's reputation within the field, in your experience?
I think it's a bit like asking, "if Bob shaved his hair and grew a beard, would it help or hurt his reputation"? Certainly you could imagine ways it would do either - many of them would depend highly on circumstances. But in my experience, academics today rarely consider spouses or beards as significant factors when evaluating someone's career. So I would expect that on the whole, they have very little effect, unless perhaps the person in question blatantly disrespects social norms.