Two separate roles with almost no overlap. The closer, sometimes called an escrow officer or settlement agent depending on the state, works for the title or escrow company. They hold earnest money, clear title issues, prepare the settlement statement, disburse funds, and get documents recorded. Their duty runs to the transaction and to their own company's requirements, and they handle the segment of the deal around the closing table.
The transaction coordinator works for the agent or investor who pays them, and covers the whole stretch from contract signature to closing. They track contingency and inspection deadlines, collect and deliver the disclosures required in that state, keep the file complete for the broker's compliance review, and stay on top of the lender, the inspector, and the title company. The closer will tell you what's missing from their file. Nobody at the title company is watching your inspection deadline for you.
Whose job it is to do what varies by state, since some states close through attorneys and some through title or escrow companies, and the split of duties shifts along with that. Ask your title office directly what they will and won't chase.
The piece people miss is that the TC also isn't the buyer's agent. They don't advise you on price, terms, or whether to walk. If your question is "should I accept this repair credit," that goes to your agent or an attorney. If your question is "was the addendum signed by everyone and delivered before the deadline," that's exactly what a coordinator exists for.