The US House just adopted a bill that would establish four income tax brackets. The US Senate just adopted a bill that would establish seven income tax brackets. A conference committee will have to reconcile that difference. What are the arguments for either position?
Two things for reference to begin: the US currently has 7 tax brackets, and you are not taxed at your highest possible tax rate for all your income - if you make $50,000 you pay 10% taxes on the first $9,325, 15% on everything between $9,326 and $37,950, and 25% on the rest (source for brackets)
The arguments for more tax brackets is that they allow for a more progressive tax structure. Historically there were far more tax brackets. There were 20-30 tax brackets for most of the 20th century which allowed the somewhat-rich to be taxed less than the super-rich (Reagan changed this). If more tax brackets were added today they would presumably be added back into these same upper ranges, as the majority (about 70%) of Americans make less than $50K which is encompassed by the first three tax brackets.
There are two different general arguments for less tax brackets, one practical and one ideological. The practical argument is that it would simplify taxes, which is technically true but the mess of taxes comes from calculating tax credits, deductions, and other facets of the tax code. Calculating your tax bill based on brackets after you have your income calculated is high-school math.
The ideological argument for less tax brackets is that people shouldn't be taxed more for making more money. The proponents and opponents of this argument are obvious, but I believe that's outside the scope of this question.
To answer your question directly in this specific context, the House wants to reduce the number of brackets while the Senate does not want to touch this particular topic. The House is more polarized than the Senate and thus the conservative writers of the bill are on average more radical than their senate counterparts, so a more radically conservative bill would be unsurprising. I cannot speak as to why the Senate has elected not to change the number of tax brackets in their bill, but seeing as its not really a contributor to the actual complexity of the tax code they purport to be tackling they may be of the mentality "if it ain't broke don't fix it".