Darkwallet uses a form of stealth addresses that is similar to Monero. This won't work with regular Bitcoin users though, you will have to use regular address unless both parties are using Darkwallet (or some wallet using a compatible stealth scheme). In a sense this feature makes Darkwallet a bit like an altcoin that is sharing the Bitcoin blockchain. If you are exchanging with regular Bitcoin users you will lose the unlinkability benefit of stealth addresses (but you can still do it and you can keep unlinkability if you never reuse addresses).

Darkwallet uses CoinJoin style mixing. This has issues that have been widely discussed elsewhere, especially when there aren't a large number of people wanting to mix at the exact same time. (There is a rendezvous process in CoinJoin where people making truncations at the same time are mixed together.) This can be still be used even with regular Bitcoin addresses, but only if the sender is using Darkwallet. A Darkwallet user who receives coins from a non-Darkwallet user will receive non-mixed coins, and also will potentially have address linkability if the address is reused. By contrast Monero uses ring signatures for mixing which doesn't require a rendezvous process the way CoinJoin does, but does have some other issues (some discussed in MRL-0001)

Since Darkwallet is based on Bitcoin it shares the rest of Bitcoins advantages and disadvantages (biggest advantage being its relatively wide adoption, but again when interacting with non-DW users your feature set is reduced). Monero is a new code base that does some things differently (for example not having a fixed block size), which again has advantages and disadvantages. The relative immaturity of the code is a big disadvantage.