Let us define a power Diophantine equation by 2 algebraic functions $f,g$ *(having different degree)* and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $l$ such that-
there are infinite integer solutions for the equation $f(x)+l=g(y)$. One may add more functions of different variable, and alter sign in the equation.


Do such power Diophantine equations exist? What are examples of such power Diophantine equations?


For example, Tijdeman's theorem states that there are at most a finite number of consecutive powers. If we write that, $x^m=y^n+1,$ then $f(x)=x^m, g(y)=y^n, k=1$, though it is not known, for $k>1$. So this doesn't fit the Diophantine Equations I am looking for. 


If the post requires more specification, please let me know.



Different solution of power Diophantine equation based on constant term
I'll assume that by "algebraic functions" is meant polynomials with integer coefficients. If both $f$ and $g$ have degree at least two, and at least one of them has degree at least three, then, trivial cases aside, it's well-known that the equation has only finitely many integer solutions. 


If both polynomials have degree two, then there is elementary theory available to decide whether there are finitely many or infinitely many solutions – this can better be discussed elsewhere, e.g., math.stackexchange.com (where it probably has already been discussed, repeatedly). 


If both polynomials have degree at most two, and at least one has degree less than two, then it's even easier to decide the number of solutions, and again math.stackexchange.com would be the place to ask (or to look to see whether there's an answer there already). 


