Thanks for the pics. Is the the labled pic the existing controller or the new one that you want to use? If existing, then it may be a challenge to find a controller with the support of headlights, tail lights, brake lights, but that just takes more research.
For the battery, please note that battery ratings will describe ##A and ##Ah ratings, as well as the voltage. 25A is the current delivering capability, which means how much current the battery can deliver 100% of the time without it overheating or being damaged. 25Ah is the current capacity of the battery over time, similar to the total energy stored in the battery. So, a 25Ah battery can deliver 25A for an hour, or 12.5A for 2 hours, or 5A for 5 hours, all of which result in 25Ah of energy used.
For your question on controller compatibility, you need to know the current rating of the 15Ah battery, which is determined by the quality of the cells used in the pack, but unfortunately, the label on your battery doesn't provide that info, but you can make some reasonable guesses.
The label states 13S6P, 48V nominal, 15Ah capacity, and 720Wh capacity, 48x15 = 720, so that makes sense; and the model number if the pack implies the use of 18650 lithium cells.
6P is the number of parallel groups making up the pack, so 6, times the discharge rating of the cells used in the pack will determine the ##A current delivering capability. We can also use the 15Ah rating to determine the current storage of the individual cells is, or 15Ah/6 = 2.5Ah (2500mAh per cell). If the pack doesn't state Samsung, LG, Panasonic, etc., then it's make of cheap generic Chinese cells 99% of the time, and a lot of the cheap packs are made of 2500Ah generic cells, so that ties to the label as well. On my cheap generic cell pack, I could reasonably pull 3A-4A per cell, and 5A had a lot of sag, so 3A or 4A may be reasonable estimate. Since you have 6P, that's (6x4) 24A, so likely the pack can be used with your 25A controller, if it's it good working condition.