The 10th Doctor was able to erase Donna's memory of him just by touching her head. Why did he need a device to do exactly the same thing to Clara?
Using the device makes the process painless
In his conversation with Ashildr, the Doctor makes clear that he could wipe Clara's memory telepathically, but suggests that using the device will make the process painless.
DOCTOR: I'm taking her back to Earth. Somewhere safe, somewhere out of the way. I'm going to wipe her memory of every last detail of me.
DOCTOR [OC]: It'll be like our friendship never happened.
ASHILDR [OC]: That may not be what she wants.
DOCTOR [OC]: 'I've done it before. Usually, I do it telepathically, but this time, I've got something better.
DOCTOR: It's quite painless.
Season 9, Episode 12: Hell Bent. Transcript from Chakoteya.net
That suggests that wiping Donna's memory was painful. The scene in Journey's End where the Doctor wipes Donna's memory certainly shows that she was in distress. It was hard to tell just from that episode if that was due in part to pain or due only to that fact that she didn't want to change back to her old self. The conversation with Ashildr suggests that there was pain.
I suspect the real answer is just "because there's no other way for the scripwriter to make it so the Doctor's memories get wiped instead." The device is merely a Macguffin to set that up.
If you want an in-universe explanation, maybe because this Doctor likes to improve upon things, e.g. sunglasses instead of screwdriver?
Remember that Donna isn't just getting a memory wipe, she needs the memories suppressed to survive
The Doctor explains that the human brain cannot take in the Time Lord mentality: if she continues in her current state, she will die as her mind will burn up. In tears, Donna protests that she wanted to continue her adventures with the Doctor as "DoctorDonna" and was willing to spend the rest of her life with him. Saddened, the Doctor says that he is so sorry; Donna then realises what he is about to do and begs him not to send her back. The Doctor then tells her that they had the best of times. Ignoring her pleas, the Doctor presses his fingers on Donna's head, wiping her mind of all her encounters with the Doctor, rendering her unconscious as a result.
By contrast, The Doctor is trying to hide Clara from her fate
DOCTOR: There was only one way to keep Clara safe. I had to wipe some of her memory.
There's also the fact that sometimes The Doctor does things that seem unintentional when that was his plan all along. I wouldn't put it past him that he let Clara wipe his memories by using the device instead of telepathy.
He didn't delete Donna's memories
Donna didn't just have memories of her time with the Doctor, her mind was infused with Time Lord knowledge. The Doctor took the knowledge and the memories, and enclosed them within her mind, cutting them off.
There was a chance that Donna could regain her memory; which she does when the Master returns. When that happens, the Doctors defence mechanism activates.
With Clara, he wants to completely erase the memories. But, its not her memories he erases - it's his. The whole conversation about erasing Clara's memories is a red herring. All along the Doctor planned to erase his memory of her, to protect her and save her life. For that, he needed the device.