Lengthen charge times with batteries
Except in very pathological cases, the replacement will let your project last 5 times as long. Consider: Let's say your project draws 1 amp. By definition, your battery will produce that current for 1 hour. That's what 1000 mAh means - 1000 mA x 1 hour. By the same token, 5000 mAh / 1000 mA gives 5 hours. 

Like I say, there are some exceptions, but the general principle is clear.

An example of an exception: Your project heats up (and keeps getting hotter) and just barely survives the one hour duration the first battery produces. With a larger battery, the project catches fire after 2 hours. Like I say, pretty unlikely, but not impossible.