Since this is a contractual dispute I imagine this is a civil matter.
As far as I know, in a typical civil matter between two parties their
power to obtain evidence from one another is more limited than a
state's power when it comes to criminal matters. For example, the
state can issue search warrants to get access to otherwise private
material.
If a state is involved in a civil matter like a contractual dispute
can it use all its criminal justice system powers or will they be
limited to powers the civilian party has.
There are differences between criminal procedure and civil procedure for obtaining evidence. This said, the differences are smaller than you might think. In a civil matter, subpoena power, or the equivalent such as a court order directing a party to disclose something specific, can be used to compel the disclosure of documents and the compulsory testimony of witnesses under oath.
Generally speaking, the fact that the information is private in the sense that it is held on private property to which the public does not have access, or that the information is subject to contractual non-disclosure agreements is not enough to overcome a court's power to compel the disclosure of information.
In some respects, the compulsory power to obtain evidence can be greater in a civil matter than in a criminal matter, because the concerns about involuntary self-incrimination are not the same in a civil matter.
In exceptional circumstances, civil courts can authorize something similar to a search warrant in a criminal case, although this is the rare exception rather than the norm.
Also, there is middle ground between true civil matters, in which the government is just one more party to civil litigation, and criminal matters. While it may not apply to a contract dispute, in many cases the government engages with private parties in an administrative process, such as the process used in tax collection, in environmental law monitoring of licensed polluters, the process of land use regulation approvals, or the process used in other regulated industries. In administrative matters, the scope of what is litigated is much more narrow and the process is tailored to a particular type of transaction, often with the private party commencing things by sending an application or some other sort of form to a government agency, and the administrative process governing the give and take between the government and the private party that leads to a final resolution.
Often, in civil litigation, the government will have information obtained in its administrative processes that may be used in its civil litigation as well as evidence (although some information provided in administrative processes cannot be used in this way due to privacy legislation).
On the same vein, may they use public prosecutors or must they hire
private representation?
Neither.
Public prosecutors are typically a distinct corps of lawyers charged solely with enforcing the criminal laws (especially in Europe where their professional certification is often distinct from that of litigators in civil practice, with England and Wales as a notable exception where a common pool of barristers are used by the government and private parties in serious litigation).
But since governments use such a high volume of legal services and frequently engage in litigation, typically governments have a large number of full time employee lawyers on their payroll who represent it in non-criminal matters.
Now and then a government (especially a small local government) may outsource its legal work or a particular legal task to outside counsel in the private sector or from another governmental entity, but this would very much be the exception rather than the norm.
Are there any distinctions between a state appearing in a civil court
and criminal court?
Yes. There are many distinctions.
In a criminal court, the state is always the prosecution, the private parties are always the defendants, and while there are affirmative defenses that can be raised by defendants, there are no counterclaims that can be asserted by the government. The government has a higher burden of proof and less of an ability to compel cooperation from a defendant. As noted before, the profession of the government lawyers involved is often different. And, in criminal cases, law enforcement officials in the executive branch and prosecutors (who are sometimes bureaucratically in the executive branch and sometimes in the judicial branch) work hand in hand together, with the government lawyers supervising the effort or simply confining themselves to the courtroom part of the process. Prosecutors usually have a higher ethical responsibility to "do justice" rather than acting purely as an advocate for the government, than government lawyers in civil matters. In Europe, serious criminal cases are also often resolved by a panel composed of professional and lay judges, while civil cases are generally handled solely by professional judges.
In civil cases, the government lawyers will typically play a bigger part in developing and managing the case and the evidence in support of it, and will be working hand in hand primarily with non-law enforcement officers. The parties are on something closer to equal footing, which isn't to say that there aren't legal doctrines specific to government litigants that favor them.
For example, in U.S. contract litigation involving a governmental party, the opposing party must prove that the government actually had full legal authority to enter into the contract, while in the case of a private party, the opposing party must only show that the person signing the contract had apparent authority to enter into that contract (with the result that contracts of private parties are almost never invalidated as being ultra vires, while this does happen now and then in the case of governmental contracts).
Europe has a variety of legal doctrines not shared by the U.S., but like the U.S. has legal doctrines in civil litigation that are one-sided in favoring the government.
