What you are doing is neither one. You are simply watching to make sure you don't overdraw, which itself suggests you might be living hand to mouth and not saving. Keeping track of your money and budgeting are useful tools which help people get on top of their money. Which tends to have the effect of allowing you to save.
How much did you spend on groceries last month? Eating out? Gas? If you were "keeping track of your money", you could say immediately what you spent, and whether that is above or below average, and why.
How much do you plan to spend in the next 3 months on gas, groceries, eating out? If you knew the answer to that question, then you would have a "budget".
And if those months go by, and your budget proves to have been accurate, or educates you as to what went wrong so you can learn and fix it... then your budget is a functioning document that is helping you master your money.
Certainly the more powerful of the two is the "keeping track", or accounting of what has happened to you so far. It's important that you keep track of every penny without letting stuff "slip through the cracks". Here you can use proper accounting techniques and maybe accounting software, just like businesses do where they reconcile their accounting against their bank statements and wallet cash.
I shortcut that a little. I buy gift cards for McDonalds, Panera, Starbucks, etc. and buy my meals with those. That way, I only have one transaction to log,
$40 - McDonalds gift card instead of a dozen little meals. It works perfectly fine since I know all that money went to fast food. A little more dangerous is that I treat wallet cash the same way, logging say two monthly entries of
$100 to cash rather than 50 little transactions of
left $1 tip at restaurant. This only works because cash is a tiny part of my overall expenditures - not worth accounting. If it added up to a significant part, I'd want accounting on that.