I recently got a Dark Brotherhood Initate follower, I was wondering if there were benefits to having her rather than a regular follower. Like does she have increased sneak and what not?

They are some of the best sneak archers in the game, Aela is a tiny fraction better at archery but a tiny bit worse at sneaking. They are also decent at melee and cannot be killed.

LOL yea that's what I thought! Never used them in my first play through, but since I'm playing an assassin type with 100 in sneak and archery right now, I thought I give them a try.
OMG what a pain! They are worse than Lydia, I swear! They stay sooooooooooo close to you, that they push you around at times. I'm like BACK OFF or you will serve Sithis! So, on sneak missions, I rather do them myself.

I made some legendary daedric daggers and bows for them. Now, she was able to handle herself pretty well in a straight up fight with some Draugr Deathlords and Draugr Wights. I then took her to a Falmer cave, trying to test sneak. She stays silent as long as I'm hidden. If the enemy just slightly spots me, she rushes at them.

But the most annoying thing was the pushing and the closeness and blocking my path at times. After these 2 dungeon crawls I sent her straight back to the Sanctuary (where she will stay FOREVER). lol

I'm finding the same issue with her getting so up close and personal that she's pushing me when I'm trying to stay hidden.
At what point do the initiates become actual members of the guild, are they any better after that?

So far my best follower has been the one I picked up at the Azura shrine. I'm mostly a Mage, so she was awesome at staying back nd not getting in my way but also I'd turn around and she would have taken down like 3 guys before I equip my spells. She was also awesome at sneaking. Maybe I need her back!

I have never seen them as anything but "initiates" that's what they'll stay.

My Nord, my first character, she's a dual wielding warrior and she just rushes everything with brute force, Vorstag (Silver Blood Inn in Markarth) is her swords and shield tank in heavy armor. As magic companions I really like Belrand (Winking Skeever in Solitude), because he can dual wield staves and Marcurio (Bee and Barb in Riften). J'zargo from the Mages College is also a great magic follower.

Now, since you are a mage, you should go with a Tank, sword and shield, who can rush at the enemies in close melee combat and you can cast your spells from a distance or concentrate on a 2nd or third target. Did you get the staff Sanguine Rose yet? That's my tank in tight situations! LOL But usually I just pick targets off with my bow now, they drop dead before they know what hit them hahaha

Oh the priestess at the Azura Shrine can be a follower? Well, this time around I got the Black Star LOL

I LOVE my sanguine rose staff! Although I've been wondering if using it counts towards conjuration

I never use the Sanguine Rose staff as my character is not a mage. I treat some daedric items as collectibles and typically take them back to my pad in either Windhelm or Solitude and store them away or mount them on a weapon rack or something aestheticall pleasing...