I have heard that Wenge can cause an allergic reaction akin to poison ivy
Not sure what you read or where, but you wouldn't expect this upon first exposure. This kind of reaction would be for people with either a built up sensitivity to wenge or with an existing sensitivity. It's little to worry about unless you fall into either of these two camps.
You do want to take precautions about the wenge dust however. Breathing protection is considered a must, and given it is a known irritant I would urge you to be paranoid about it and wear at least a good dust mask every time you work the wood, even if it's just for a couple of minutes. This would be most particularly when using power tools and when sanding. Worth carefully cleaning up the dust afterwards too (while still wearing the dust mask).
I am also wondering which wood would be easier to work with
Well purpleheart is significantly harder than wenge, and many people find it difficult to work with as a result1 but it depends on how you intend to work it and the sharpness of your edges how much difficulty you can expect.
If you're used to working with harder woods you should have few surprises.
If you're used to working in pine however their hardness might come as a bit of a shock! They'll also be much more splintery than you're used to. Sharp edges are a must and expect jobs to take longer. You may even find you need to stop more frequently to hone edges due to their blunting effect.
and which will finish better
That's really up to you more than anything. They're both hard, dense hardwoods so inherently will finish well.
I suppose the real issue in either case is whether the open grain structure is something undesirable for the piece in question, but that's easily tackled with grain filler so warnings about this making finishing difficulty are basically an irrelevance IMO — grain fill, problem solved.
Regardless of grain filling there are users of both wenge and purpleheart who are more than happy with the way they look after they've finished them and if they can do it it's likely you can too.
I am trying to achieve a very dark finish
If that's all you require then there's no real need to use either of these woods2. A dark colour, all the way to black, can be achieved on any wood with staining and/or the use of a coloured overcoat.
You could make the job a lot easier for yourself by using something softer and simply colouring it dark. Hard maple/sugar maple seems like a good candidate, as would black walnut, but you could even start with poplar if it's not too soft for you.
1 FWIW I've seen the opinion expressed that "it's not a beginner wood")
2 If you want to push yourself and have a bit of a challenge then by all means use either wenge or purpleheart for this, but if that's not one of the things you're looking for here then save yourself some hassle and pick something softer.
