There are quite a few reasons why this may not work: firstly, do not concatenate strings to build a SQL command. It leaves you wide open to accidental or deliberate SQL Injection attack which can destroy your entire database. Use Parametrized queries instead. This may cure your problem without additional work.
string ConnectionString = @"Data Source=PHIL-PC\SQLEXPRESS;Initial Catalog=Sudoku;Integrated Security=True";
SqlConnection conn = new SqlConnection(ConnectionString);
conn.Open();
string namesub = nameTextBox.Text;
string emailsub= nameTextBox.Text;
string SQL1 = "insert into player(Player_name,Player_email) values (@NM, @EM)";
SqlCommand cmd1 = new SqlCommand(SQL1, conn);
cmd1.Parameters.AddWithValue("@NM", namesub);
cmd1.Parameters.AddWithValue("@EM", emailsub);
cmd1.ExecuteNonQuery();
But it would be worth checking if you really wanted the name in both the name and email fields...