And here's the code for it in case anyone needs it for future. In my case, the field names were identical, but if you have a number of different fields, it'll be good to use custom settings for mapping the field names and then using describes as here to get the same.

public with sharing class ContractCreator {
	Map<String, String> oppRecordTypes = new Map<String, String>{};
	Map<String, String> contractRecordTypes = new Map<String, String>{};
	public ContractCreator(){
		//Create a map between the Record Type Name and Ids for retrieval later
		String[] objNames = new String[]{'Opportunity', 'Contract'};
		List<RecordType> rtypes = [Select Name, Id, sObjectType From RecordType where isActive = true AND sObjectType IN :objNames];
	    	for(RecordType rt: rtypes){
	    		if(rt.sObjectType == 'Opportunity')
        			oppRecordTypes.put(rt.Id, rt.Name);
    			else
    				contractRecordTypes.put(rt.Name, rt.Id);
	    	}
	}
		
	public void createContracts(Map<Id, Opportunity> opps){
		List<Contract> contractsToCreate = new List<Contract>();
		//Get all the common fields in the two objects
		Set<String> fieldsToPopulate = new Set<String>();
		Map<String, Schema.SObjectField> contractSchemaMap = Schema.SObjectType.Contract.fields.getMap();
		Set<String> oppFields = Schema.SObjectType.Opportunity.fields.getMap().keySet();
		for(String cField: contractSchemaMap.keySet()){
			for(String oField: oppFields){
		 		if(cField == oField){
		  			fieldsToPopulate.add(cField);
		 		}
			}
		}
		for(Opportunity opp: opps.values()){
			SObject contr = new Contract();
			for(String s: fieldsToPopulate){
				Schema.DescribeFieldResult contractField = contractSchemaMap.get(s).getDescribe();
				//Populate all the non-null fields if they are createable
				if(opp.get(s) != null && contractField.isCreateable()){
					System.debug('Populating..: ' + s +' with '+opp.get(s));
					contr.put(s, opp.get(s));
				}
			}
			Contract cnt = (Contract)contr;
			cnt.recordTypeId = contractRecordTypes.get(oppRecordTypes.get(opp.recordTypeId));
			cnt.Organisation_Name__c 	= opp.AccountId;
        		cnt.Opportunity__c 		= opp.Id;
        		cnt.Contract_Name__c 		= opp.Name;
	        	
			contractsToCreate.add(cnt);
		}
		insert contractsToCreate;
	}
}