Hashi's is an autoimmune thyroid disease. When you have Hashi's, your immune system is producing antibodies that attack the thyroid, eventually causing hypothyroidism.
"Thydroid deficiency" is also hypothyroidism, but from a cause other than autoimmune or an indeterminate cause.
There can be side effects from levothyroxine. By and large, it's well tolerated since it's not a medicine, per se, but a replacement of the hormone your thyroid would produce if it could.
I totally agree with goolaraa.
I have had no side effects from my T4 medication.
My Endo told me when starting out on Synthroid and also each time he raises my dose that I may have some palpitations or "heart flutters." Each person is different. I have noticed them a bit, but nothing terrible. He said it should go away within a week or two after in your case, starting the med...in mine, when he raises it. Nothing to be alarmed about.