We are testing whether our actual data is the same as our projected data.
No you're not. You can answer that at a glance (the data are different). Hypothesis tests are for doing something else.
It's not clear however, that a hypothesis test is a good choice for your problem, and even if it were, I wouldn't use these.
We had been using the t-test until we realized the data might not be normal and we are using ~12 observations per sample. Therefore I started using the Mann Whitney test.
I think a big problem here is that actual-vs-projected should be paired (by the observation that they're the actual and projected of). Consider that if you ignore the pairing that you'd regard both these as equally adequate:
Actual: 6 9 12 15 16 14 19 20 20 22
Projected 1: 11 12 13 14 15 16 17 18 19 20
Projected 2: 20 19 18 17 16 15 14 13 12 11
However, even taking account of pairing on observation doesn't solve the underlying problem
-- that projections may be exactly right on average, but still be terrible. You want the individual projecteds somewhere close to individual actuals (close overall) and a test of means doesn't address that at all.
A better way to measure difference between actual and projected would be something like a sum of squares of error but I don't think you should be trying to test anything. It doesn't really answer the kind of question that it makes much sense to ask (such as "are our projections good enough for our purposes?" or "is this projection approach performing better than that one?" - those kind of questions make sense but aren't hypothesis-testing questions).
I have read some places that the test is not appropriate unless the variances are equal,
It depends on why you're using it. For some uses, you need even more than equality of variance, you need identical shape. For other uses it doesn't matter.
and other places have said you can still use it if you don't care if the shapes are identical.
Basically, that's correct.
The Mann-Whitney U statistic (when scaled by the number of comparisons) estimates the probability that a random observation form one population exceeds a random observation in a second one, irrespective of whether the two are similar in shape.
So it's possible to get a U statistic that shows no indication of a difference (in the sense of being close to what you expect if the null were true) when two distributions are nothing alike.
if you make the additional assumption that the shapes are the same, apart from a possible shift, then the Mann-Whitney is a location test (and quite a good one in many situations).
So do I need to test each sample against the other to see if the variances are equal?
That depends, but what you should do is consider whether a hypothesis test is even the most appropriate approach for your problem (I really don't think it is), and if so, then think about the pairing issue. Those should come first.
Do the interpretation of the results differ if the variances are not equal?
Possibly. If the shapes are the same, you can say something more specific.