They're trying to get rid of your claim.
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They are technically two different types of motions, so it depends on the context in which you are asking. A "motion to dismiss", seeks a court order to dismiss a plaintiff's claim on various statutory grounds (ie- "fails to state a cause of action..."; claims were decided in prior litigation [res judicata and/or collateral estoppel]; claim untimely brought [statute of limitations]; etc. A "motion to quash" generally seeks a court order to "quash" [to suppress or to void] something - typically a subpoena, Notice, Demand, etc., served by an adversary that seeks to compel the production of a person or thing. One does not "quash" a lawsuit; one seeks to have a lawsuit "dismissed".
I am not sure if you are the plaintiff or the defendant. If you are the defendant turn the claim over to your homeowner's insurance company to defend it. If you are the plaintiff, be forewarned that these claims are seldom successful even if your facts are "good."
In answer the question regarding the technical difference between the two motions: A motion to dismiss seeks to eliminate the claim for any reason. It could be something technical and curable, like not attaching a document to the Complaint; something substantive, but potentially curable like not alleging the elements of slander; something technical but not curable, like the court has no jurisdiction over the defendant; or something substantive and not curable, like the statute of limitations has run.
A motion to quash service of process (or process) is an attack on either the way the Complaint and Summons are worded, issued, or served, and/or on the jurisdiction of the court over the defendant. If it is well-taken it might be curable or not. It will lead to a dismissal, with leave to amend if it is curable, or without leave to amend if it is not curable.
This is a summary based on incomplete facts. You should not rely on it as legal advise. No attorney-client relationship is intended to be formed. You may call me 772-562-4570; email me firstname.lastname@example.org, or visit my website http://www.millerlawoffices.us