"Like" is a preposition, and should not be used to introduce a verbal phrase. The conjunctive use of "like" (it seems like + verbal phrase) is a casualism which has crept into the lower tiers of standard English and become increasingly common in colloquial prose, but which is nevertheless still regarded as incorrect by most prescriptive standards.
"That" is a fully standard, register-neutral (neither formal nor informal) conjunction.
On a different note, "seem like" can introduce a noun, whereas "seem that" can only introduce a clause (verbal phrase). You can't use "that" in the first of the following sentences:
"He seems like a nice guy".
"It seems that he's a nice guy".
You also wouldn't normally use "seem that" if the subject of the sentence is a person. "He seems that he's trying to open the door" is an awkward sentence when compared to more natural-sounding options:
"It seems that he's trying to open the door."
"He seems to be trying to open the door."
"He seems like he's trying to open the door."
On a final note, I'd be more inclined to use "seem like" more often when drawing comparisons and "seem that" when stating a perceived fact.
"It seemed like (it felt as if / it felt as though) the walls were closing in around me". (The room was cramped and I was feeling suffocated).
"It seemed that the walls were closing in around me". (I noticed the walls were moving closer).
This is a subtle differentiation based on the equivalence between "like" and "as if / as though". Actual usage might not bear it out.