If you compare the passmark scores the 6300 scores better, just because a processor has a higher clock speed does not make it faster.
I also would not recommend upgrading from a dead platform to another dead platform (AM3+)
My recommendation for a 100 dollar processor is an i3 6100
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($110.99 @ SuperBiiz)
Total: $110.99
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-08-03 19:25 EDT-0400
However, that would require a new motherboard and RAM, which is over budget, but leagues ahead of both processors.