Relation between linear utility function and U=max{x,y}
The optimal choice set for a max function and a perfect substitutes function with equal relative prices share some solutions [i.e, boundary solutions], but in general, the indifference curves, and hence non-boundary solutions, are different.
Main Idea
For both a max(x1 x2) and perfect_sub(x1 x2) utility function, the point, say, m/p1 (or m/p2) would maximize utility. So both utility functions share boundary solutions. But think about the IC of m/p1 for a perf_subs consumer and think about the IC of m/p1 for a max consumer. You see the points that the two consumers think are 'just as good' as that m/p1 point are very different.
That is, the same boundary bundle might be a solution to both utility functions, but the other solutions will differ.
Logic
The reason is that U = perfect_subs is a (not-strictly) convex utility function, whereas U = max isn't. That is: the consumer is either indifferent between, or actually prefers, less extreme combinations to more extreme combinations for the former. As for the latter? Well, they just care about the max; they like extreme combinations (e.g, [C = (m/p1, 0)])
That's why for U = perfect_subs with relative prices the same, both A = (100,0) and B = (ß(x1), (1-ß)x2) have the same utility. That is,B lies on a straight line IC connecting (100, 0) to (0, 100). Whereas for U = max(x1 x2), B would not be on the same IC as A (unless ß is 0 or 1, in which case, we're back to talking about boundary solutions!)
The word, max is confusing here. Essentially, the perfect_subs (x1 x2) consumer wants to maximize total goods. The max(x1 x2) consumer just cares about maximizing the larger quantity in his {x1, x2} bundle.
Tl;DR
To be more concrete: If A = (100,0) and B = (75, 25), a perf_subs consumer is indifferent between A and B; a max consumer is not.