Rest between sets is a vital part of your workout and can have a direct correlation to whether or not you achieving your goals in a certain time. Experts recommend 30 seconds to 2 minutes rest is adequate for non-strength gaining exercises. However, strength training requires much more rest - at least 3 minutes during your warm-up sets, and 4-10 minutes required for barbell training sets. Competitive strength and power lifters often rest 10 minutes or more.

Does the amount of time that you rest in between reps affect your results?

YES. In a landmark study published in 1997 by W.J. Kraemer at Ohio State University, athletes completed more reps in 3 sets after resting for 3 minutes compared to when they rested for only 1 minute. A similar, earlier study showed a 7% increase in squat strength after 5 weeks of training with 3 minute rest periods. Athletes who rested for just 30 seconds only saw a 2% increase in their squat. In fact, a lack of adequate rest during strength training can lead to some major muscle building roadblocks.

1. During set recovery, lack of rest can lead to a decrease in competitive drive, determination, and training intensity. 

2. Lack of rest can inhibit the physiological tools our bodies use to adapt to the stress of strength training. 

The amount of rest required between warm-up sets should be about the time it takes to load the bar with the next warm-up set. And you don't want to do so many warm-up sets that the warm-up becomes your workout. Spacing it out with 4 - 5 smaller sets and decreasing the number of reps as you get higher and closer to your work set weight is a great way to negate that possibility. And, as you start to lift heavier work sets, your warm-up will eventually require a little more rest between sets. 

The optimal time to rest between sets is dependent on a variety of factors: intensity of the set, fatigue and nutritional status of the lifter, their age, temperature of the facility, and injury status if any. The optimal rest time for barbell and strength training usually can be determined by whether or not failure happens inside of 4 months.

One 2008 study found that the longer rest times could be adjusted slightly after initial volume goals were met to focus on other conditioning priorities - but athletes had to start with longer rest times. Which emphasizes my point, that if you fail within the first 4 months of starting a strength training program, you might not be resting enough. After 4 months, the weight should be heavy enough and your body awareness should be such that your mind will let you know (either fear or excitement - identical physiological responses) if you are rested and ready to begin the next set. 

Sources:

Practical Programming for Strength Training 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3899651/

https://www.researchgate.net/publication/232198562_A_Series_of_Studies-The_Physiological_Basis_for_Strength_Training_in_American_Football_Fact_Over_Philosophy