Bitcoin core can also be run in prune mode... That way you only need about 3 Gb diskspace. You still need to sync completely, but the blocks get pruned afterwards. core 0.13.0 is deterministic, the previous versions weren't. Advanced coin controll features are there, but some might have to be used using the console.
I assume you're talking about multibit HD: this is a deterministic SPV wallet, like shield132 mentioned, it doesn't need to sync completely (only the headers), making it smaller and faster (especially initially). personally, i find multibit HD lacking advanced coin controll features, but i must say it's easy to use, and pretty to look at
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AFAIK, both are safe to use, as long as you backup the seed (or wallet.dat), keep your pc malware free and chose a strong password.
You must realise you're not paying the transaction fee to the maker of the wallet, it's a fee that is payed to make sure the miners have an incentive to add your transaction to a block. So, it might not be wise to pick the wallet that lets you create transactions with the smallest fees, because those transactions might end up "hanging" or even "forgotten" by the nodes.
Personally, i prefer electrum, because it has a lot of coin controll features (like core, but i find the coin controll features of electrum more newbie-friendly, because they can be accessed via the gui), but is deterministic and SPV like multibit HD. It is said to be less secure to a brute force attack when using a weak password tough, and it's not as pretty as multibit HD.