Generally speaking, there often isn't much difference between the two. However, when there is, it's a matter of subjectivity versus objectivity.
In that respect, you can look at the two in the following way.
I have a purpose.
This means that I actually do have a purpose. Other people can look at me and see that I'm doing something for a reason. Even if I don't see it myself, others do.
I have a sense of purpose.
This means that I believe I'm doing something for a reason. It may be something that nobody else sees or cares about (and they might say I have no purpose), but I believe I'm doing something meaningful.
There is also a different shade of meaning between the two—one of enthusiasm.
Let's say you're captured by cannibals and, over a course of days, are slowly eaten. There can be no doubt that your purpose at that point is to feed the larger group.
If you are depressed by this turn of events, you might say:
My only purpose now is to be a food source.
On the other hand, if you had been depressed about life before this happened, and didn't think you were doing anything worthwhile with yourself, and are actually happy about what has transpired, you could say with enthusiasm:
I finally have a sense of purpose again—to feed these worthy people!
In the sense of version here, it's expressing a positive attitude rather than just a statement of fact.
In your own sentence, I personally find sense of to carry more meaning and be more appropriate to what I'm assuming it's trying to express. (Although I could be wrong.) As I read it, it's describing somebody without much to do and, therefore, feeling as if they aren't contributing. They may know that their actual purpose for working is to put food on the table, but they no longer have a sense of purpose. (The work itself seems monotonous and uninspiring.)
So, while it may not be wrong to omit sense of, it seems more likely that it should remain.