Salida de SQL Server y C# diferente

La mejor manera es formatear su entrada en dd/MMM formato /aaaa. Ayuda mejor a SqlServer a identificar el mes y el día. Recomiendo encarecidamente utilizar el formato MMM siempre que se comunique con SqlServer.

DateTime fromDate, toDate;
fromDate = DateTime.Parse(dtpFrom.Text);
toDate = DateTime.Parse(dtpTo.Text);

SqlCommand objCmd = new SqlCommand("SELECT CONVERT(char(80), i.InvDate,3) AS InvDate, 
i.InvoiceNo,
i.EmployerCode,
i.TaxAmount + i.SubTotal AS Amount,
'' AS Payment,
pd.GivenName 
FROM    dbo.Invoice i 
INNER JOIN dbo.PatientDetails pd ON (pd.MedicalRecordID = i.MedicalRecordID) 
WHERE datediff(d, '" + fromDate.ToString("dd/MMM/yyyy") + "', i.InvDate) >=0  
AND datediff(d, '" + toDate.ToString("dd/MMM/yyyy") + "', i.InvDate) <=0", objConn);

o si tiene alguna restricción para hacer lo anterior, convierta la entrada usando CONVERT()

SqlCommand objCmd = new SqlCommand("SELECT CONVERT(char(80), i.InvDate,3) AS InvDate, 
i.InvoiceNo,
i.EmployerCode,
i.TaxAmount + i.SubTotal AS Amount,
'' AS Payment,
pd.GivenName 
FROM    dbo.Invoice i 
INNER JOIN dbo.PatientDetails pd ON (pd.MedicalRecordID = i.MedicalRecordID) 
WHERE datediff(d, CONVERT(datetime, '" + dtpFrom.Text + "', 105 ), i.InvDate) >=0  
AND datediff(d, CONVERT(datetime, '" + dtpTo.Text + "', 105 ), i.InvDate) <=0", objConn);