Permitted vs. Required to Remove Influence

Is there a meaningful difference between the right of lawmakers to regulate campaign finance and an amendment requiring them to do so?

Answer

Yes, there is a serious and significant difference.
The difference is between the meaning of the word "may" and the word "must". Many bills to amend the constitution have been proposed that only assert that Congress "may" regulate campaign finance and spending limits. Some bills say that Congress "must" regulate campaign finance and spending.

For example, Rep. Dennis Kucinich's (OH-10) 2012 bill simply stated that no source of funding will be permitted under the law unless it is a public source. 

In 2013, Rep. Donna Edwards (MD-4), put forth a bill that simply asserts that nothing shall "prohibit" Congress from regulating money in politics. However, while nothing in this bill is stopping Congresspeople from limiting their corporate benefactors influence, nothing is requiring them to do anything at all, either.

In 2019, Rep. Pramila Jayapal (WA-7) proposed a bill that insured that no significant influence from private contributors will be permitted under the law, which is the equivalent in effect of banning all sources, and meets National Intervention's requirement of citizen-owned, not privately-owned, elections.