The speaker wire that runs from your amp to the speaker is a two conductor wire meaning that it has a side for the positive and negative terminal on the Speaker. The objective is to supply the electrical current to the terminals. To do that you need a solid contact. The existing connectors have a female terminal that conveniently slides over the male terminals of the original speaker. The original design (with two different size terminals) made it impossible to accidentally connect the positive wire to the negative terminal and vice versa. But as you have discovered, the replacement speaker has two terminals that are the same size.
I have good news. First, the new speaker does label the positive and negative terminal with a + for positive and - for negative. So you should have no trouble connecting it properly. You should note that the new speaker has the positive and negative terminals on opposite sides than the original speaker (they are flip flopped).
Second, you won't need a new speaker wire (or cable). You simply need a new connector. If there is an electronics shop or auto parts store near where you live, you can take the speaker in and they can help you find the correct connector. You will then need to cut the original connector off by cutting the wire as close to the connector as possible. If there is not enough slack left after cutting the wire, you can use a splice sold at the same electronics store, to add a new piece of wire as an extension. They make splices that you simply insert the two wires you wish to join into and clamp them with pliers. A photo of the speaker wire will aid in determining what gauge wire you need to make an extension if needed.
If you don't want to cut the speaker wire to change the connector, you could buy a male blade terminal that fits the existing connector and a female connector that fits the new speaker and clamp them on either end of a short jumper wire of the appropriate gauge. The picture below will show you the type terminals that you want. You can order them online from an electrical supply company if you can determine what size - and if there is not an electronics or auto parts store near you.
If you or your friend have access to a soldering iron, instead of using the connector, you could solder the wire to the speaker terminal, but this would make future speaker replacement slightly less convenient.
Another thing to try, it prying the connector that is too small apart carefully with perhaps a dull knife blade and some pliers. Then try to reshape it to fit the new connector and clamp it on with pliers and add some electrical tape for good measure.
Any of the suggestions above will work. All you really need to do is provide good contact between the terminal and the speaker wire to make an electrical connection and be sure it is secure enough that it does not vibrate loose inside your cabinet.