Excel – How to save xlsm as xlsx

You can use a library called ExcelLibrary. It's a free, open source library posted on Google Code:

ExcelLibrary

This looks to be a port of the PHP ExcelWriter that you mentioned above. It will not write to the new .xlsx format yet, but they are working on adding that functionality in.

It's very simple, small and easy to use. Plus it has a DataSetHelper that lets you use DataSets and DataTables to easily work with Excel data.

ExcelLibrary seems to still only work for the older Excel format (.xls files), but may be adding support in the future for newer 2007/2010 formats.

You can also use EPPlus, which works only for Excel 2007/2010 format files (.xlsx files). There's also NPOI which works with both.

There are a few known bugs with each library as noted in the comments. In all, EPPlus seems to be the best choice as time goes on. It seems to be more actively updated and documented as well.

Also, as noted by @АртёмЦарионов below, EPPlus has support for Pivot Tables and ExcelLibrary may have some support (Pivot table issue in ExcelLibrary)

Here are a couple links for quick reference:
ExcelLibrary - GNU Lesser GPL
EPPlus - GNU (LGPL) - No longer maintained
EPPlus 5 - Polyform Noncommercial - Starting May 2020
NPOI - Apache License

Here some example code for ExcelLibrary:

Here is an example taking data from a database and creating a workbook from it. Note that the ExcelLibrary code is the single line at the bottom:

//Create the data set and table
DataSet ds = new DataSet("New_DataSet");
DataTable dt = new DataTable("New_DataTable");

//Set the locale for each
ds.Locale = System.Threading.Thread.CurrentThread.CurrentCulture;
dt.Locale = System.Threading.Thread.CurrentThread.CurrentCulture;

//Open a DB connection (in this example with OleDB)
OleDbConnection con = new OleDbConnection(dbConnectionString);
con.Open();

//Create a query and fill the data table with the data from the DB
string sql = "SELECT Whatever FROM MyDBTable;";
OleDbCommand cmd = new OleDbCommand(sql, con);
OleDbDataAdapter adptr = new OleDbDataAdapter();

adptr.SelectCommand = cmd;
adptr.Fill(dt);
con.Close();

//Add the table to the data set
ds.Tables.Add(dt);

//Here's the easy part. Create the Excel worksheet from the data set
ExcelLibrary.DataSetHelper.CreateWorkbook("MyExcelFile.xls", ds);

Creating the Excel file is as easy as that. You can also manually create Excel files, but the above functionality is what really impressed me.

In order to do this you will need to use the Open XML SDK 2.0. Below is a snippet of code that worked for me when I tried it:

byte[] byteArray = File.ReadAllBytes("C:\\temp\\sa123.xltx");
using (MemoryStream stream = new MemoryStream())
{
    stream.Write(byteArray, 0, (int)byteArray.Length);
    using (SpreadsheetDocument spreadsheetDoc = SpreadsheetDocument.Open(stream, true))
    {
       // Change from template type to workbook type
       spreadsheetDoc.ChangeDocumentType(SpreadsheetDocumentType.Workbook);
    }
    File.WriteAllBytes("C:\\temp\\sa123.xlsx", stream.ToArray()); 
}

What this code does is it takes your template file and opens it into a SpreadsheetDocument object. The type of this object is Template, but since you want it as a Workbook you call the ChangeDocumentType method to change it from a Template to a Workbook. This will work since the underlying XML is the same between a .xltx and a .xlsx file and it was just the type that was causing you an issue.