You need to distinguish between events and random variables.
I think you mean to say random variable $X$ has the discrete
uniform distribution on the integers between 100 and 400 inclusive.
Then $P(X = i) = 1/301$ for any integer $i \in [100, 400].$
Also $Y is an independent random variable with the same distribution.
In order to grasp what this problem involves, I suggest you
consider rolls of two fair dice, a red one and a green one.
Write out the $6 \times 6$ array of results that represents the sample space. Then consider what the possible red minus green differences can be. Finally, count the outcomes in the corresponding diagonals of the diagram.
Your problem is to do the same thing, but for a $301 \times 301$ array. That is too big to write out, but now you know the principle
and should be able to solve the problem.
If you have R (or similar statistical software available) a simulation gives the approximate probability distribution of
the difference. I have done it below for the case of the dice so I can show the answer here.
In order to simulate your problem and roughly check your answer, you could change c from 1 to 100 and d from 6 to 400 (and round to 4 places, not 2). The printout will be very long, but perhaps instructive; the graph will show a discrete distribution with a triangular 'envelope'.
m = 10^6; c = 1; d = 6
x = sample(c:d, m, repl=T); y = sample(c:d, m, repl=T)
w = x - y; TAB = table(w)/m; round(TAB, 3)
## -5 -4 -3 -2 -1 0 1 2 3 4 5
## 0.028 0.056 0.083 0.111 0.139 0.166 0.139 0.112 0.083 0.056 0.028
plot(TAB) # plot not shown
For the dice problem, possible values from -5 through 5 (least frequent values), and the most frequent value is 0