The West German Cabinet today approved a 1982 budget that foresees substantial savings in defense spending and some social programs but fails to resolve a deep split between Bonn's ruling parties over how to combat mounting unemployment.

The political dispute, papered over for now, is expected to reemerge later in the year if West Germany's jobless rate continues to grow, as is expected. Official figures released this week put unemployment in West Germany at 1.3 million in August, or 5.5 percent of the work force, the worst August figure in 29 years.

The problem of stimulating employment while cutting public spending in the wake of an economic slump and a loss of international confidence in the West German mark has aggravated the traditional ideological gulf between the Social Democratic Party of Chancellor Helmut Schmidt and its more conservative junior coalition partner, the Free Democratic Party.

Weeks of divisive wrangling over next year's budget saw the Social Democrats, with strong labor union backing, pushing for large government expenditures to stimulate investment by private industry.

The Free Democrats favored cuts in social security spending and tax relief to encourage business investment.

The Free Democrats won. The budget, which still requires enabling legislation from the parliament, contains no major jobs creation program.

But both parties took the extraordinary step of issuing conditional statements that leave the dispute simmering.

The approved budget totals 240.8 billion marks ($99.3 billion), a 4.2 percent increase over this year. West German Finance Minister Hans Matthoefer told reporters today that the total represented a saving of $7.4 billion on earlier projections. These savings are aimed at taking pressure off West Germany's money market and thereby free more money for business investment. .

The final figure for Bonn's 1982 defense spending showed no change from the one announced in July. It will increase $700 million over last year's $16.3 billion, a rise of only 4.2 percent. Taking West Germany's projected inflation rate of 5 to 6 percent for next year into account, this represents a real decline in Bonn's defense spending.