I think you are misunderstanding how capital gains work. It's not about the difference in value between the property you're selling and the property you're receiving at the time of the exchange. It's the difference between your basis in that property (basically its "initial value"), and the value of the property you get in exchange.
We don't have enough information to correctly determine the gains in the examples you gave, because we don't know the basis. (Also, it's a little unclear whether your numbers refer to the value of the cryptocurrency per unit, or for the entire lot.) So consider instead the following sequence of transactions:
On June 1, 2015, Alice spends $100 to buy 1 bitcoin.
On May 1, 2016, Alice exchanges her 1 bitcoin for 100 monero. At that date, Monero has a fair market value of $1.50 per unit. (It's irrelevant for tax purposes what the market value of Bitcoin is on that date, but it would be very surprising if it were significantly different from $150 per unit, since that would indicate that either the market is very inefficient or Alice is not really paying market price.)
On April 1, 2017, Alice sells her 100 monero at $1.75 per unit, receiving a total of $175 in cash.
The question is whether transaction 2 is a 1031 like-kind exchange. Let's see what happens in either case.
If it's not a like-kind exchange (case A)
Alice's basis in her 1 bitcoin was $100. When she trades it for $150 worth of Monero, that's a taxable gain of $50, which she must include in her taxable income for 2016. Now her basis in her 100 Monero is $150. When she sells it in transaction 3, she has a taxable gain of $25, which she must include in her taxable income for 2017.
If it is a like-kind exchange (case B)
Alice's basis in her 1 bitcoin was $100. When she trades it in transaction 2, no taxable gain is realized, and she reports nothing on her 2016 taxes. She still has a chunk of property with a basis of $100, only now it consists of 100 Monero instead of 1 Bitcoin. When she sells it in transaction 3, she has a taxable gain of $75, which she must include in her taxable income for 2017.
Note that the total amount of taxable gain is the same in either case: $75, the difference between the amount of cash Alice initially put in and what she got out. The 1031 exchange doesn't really avoid the taxes, it only defers them. So you might ask, why does it matter?
Use of the money. In case B, Alice gets to wait an extra year before paying the taxes on $50 of her gain. So even if the same tax is owed on that $50 in either case (let's say $8 in taxes), in case B, Alice gets to sit on that $8 for an extra year. She could invest it and keep the interest. Or, if it's a hardship for her to pay the taxes - say she'll have to sell her car to pay them - in case B she gets to drive her car for an extra year.
Long-term versus short-term gains. In case A, both gains are from property Alice held for less than one year. So she pays short-term capital gains tax on both the $50 and the $25 gain, which is typically at the same rate as her regular income. But in case B, she held the property ("some cryptocurrency") for more than one year, and the $75 is taxed at the long-term capital gains tax rate, which is generally lower. So case B would result in less total tax paid.
Tax owed in different years. Alice's changing tax situation might make it better to have gains in one year than another. Maybe her other income fluctuates, putting her in a different tax bracket. Maybe she has capital losses to offset the gains in one year, and not in another.
Basis step-up. Suppose that Alice dies on March 1, 2017, when the market price of Monero is $1.74 per unit. She leaves her holdings to Bob, who decides to sell on April 1, 2017. In either case, by rule, Bob inherits the property with a basis of $174, so he has a taxable gain of $1 when he sells. In case A, Alice already paid tax on $50 of gains on May 1, 2016. In case B, Alice never owed any capital gains tax at all, so she (or her heirs) come out ahead.