RD := (S::set) -> {ListTools:-MakeUnique([S[]], 1, LinearAlgebra:-Equal)[]}:

RD( {Matrix([1])} union {Matrix([1])} union {Matrix([1]), Matrix([2])} );

fn must be declared as procedure, not function

f1:=proc(t::numeric, fn::procedure)::numeric;

sin()  and sin(x)  do not have the type procedure; their type is function in Maple.

So, sin  or sin()  are functions mathematically, but for Maple, sin is a procedure and sin() gives an error because the procedure sin cannot be called with 0 arguments.
You may check whattype('sin()');

whattype(sin);
whattype(sin(x));

                           procedure
                           function

The "interrupt current operation" button is in the bottom left corner of the screen, but it usually does not work.
The new interface has many bugs ...
Better use the "old" Screen Readers one. 

Download 2curves-vv.mw
(edited for Global max).

The constraints must be included in a set.

Optimization:-NLPSolve( TP_T(p1, p2), {C1, C2}, assume = nonnegative,  maximize );

diff( c * x / (1+x), x);

Optimization:-Maxima computes only local maxima.
In your case when computing the max in the interval {256.9131467 <= Pc, Pc <= 1436.173707},
Pc = 256.9131467  is a local maximal point. The global maximal point is of course Pc = 1436.173707.
If you want a global max (1-dimensiomal only!) you must use NLPSolve (and also method=branchandbound,   but it seems to be the default here).

When you meet such an aparent discrepancy you should look at a simple example; it is also much better to post a minimal example and with standard notations. Here is such an example:

f:=(x-2)^2:
Optimization:-Maximize(f, {1 <= x, x <= 5});
Optimization:-NLPSolve(f, x=1..5, maximize); # method=branchandbound);
plot(f, x=0..5);

                         [1., [x = 1.]]

                         [9., [x = 5.]]

C := C1 - C2 is a rational function (with 15 indeterminates). It will be enough to compute the minimal value for C and - C.
This can be obtained with the command  DirectSearch:-GlobalOptima(C, vars);
Here DirectSearch is a package which can be downloaded at https://www.maplesoft.com/Applications/Detail.aspx?id=101333

Actually, both min values are < 0, so, the answer to your question is that C1<C2 and C2<C1  are both false.
This can be seen directly (without DirectSearch) using:

P:=numer(normal(C1-C2)) / (lambda*varphi):
eval(-P, {Cm = 1, Cr = 10, Pu = 1, U = 2, a = 9, beta = 9, k = 10, lambda = 10, p1 = 1, tau0 = 10, upsilon = 10, varphi = 10, w = 10, varepsilon = 10, U[0] = 10});
#                     -63035971982554554000

eval(P, {Cm = 89, Cr = 65, Pu = 81, U = 50, a = 37, beta = 43, k = 44, lambda = 0, p1 = 85, tau0 = 74, upsilon = 75, varphi = 59, w = 62, varepsilon = 48, U[0] = 41});
#                       -19372211411803488

Download LM-vv.mw

Maple cannot compute the limit, but your expression 
Sum(Sum(j/(j^2 + k^2), j = 1..n), k = 1..n) / n
is a Riemann sum of the double integral

int(y/(x^2+y^2), [x=0..1,y=0..1]);

and this is the result of the limit (not difficult to justify).

Download test12-vv.mw

B:=(4*exp(x/2) - 3*x - 2*exp(x/3)+1)^11 + (4*exp(x/2) + x - 2*exp(x/3)+1)^11 - 13:
A:=combine(expand(B)):
A1:=simplify(A, size): B1:=simplify(B, size):
length~([A, B, A1, B1]);
MmaTranslator:-Mma:-LeafCount~([A, B, A1, B1]);                   

                    [6209, 120, 3487, 120]

                      [1539, 38, 976, 38]

It will be very difficult (no algorithms available) to simplify A to B.  For the other software too!

eval(A-B, x=2*t):
simplify(factor(expand(%))) assuming real;

                     0

Maple 2024 does not solve the problem.
Here is a workaround:

convert(series((f:=(lhs-rhs)(eq)),t,16),polynom):
solve([coeffs(%,t)]);
eval(f,%); #check

        {A[1]=0, A[2]=0, A[3]=-7/25, A[4]=1/25, A[5]=-2/5, A[6]=1/5, A[7]=3/16, A[8]=0, A[9]=0, A[10]=3/8}
        0